#498573

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	#576
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

642 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2,4,3)(5,6,10,15,11,34,27,46,13,18,26,42,12,17,8,24)(7,19,20,38,35,47,56,40,23,48,49,39,22,36,33,14)(9,16,29,41,30,60,55,68,32,45,50,63,31,44,25,53)(21,43,28,57,54,66,61,65,52,67,59,64,51,62,58,37);(1,3,4,2)(5,7,13,23)(6,14,18,40)(8,20,27,49)(9,28,32,59)(10,33,26,56)(11,22,12,35)(15,36,42,47)(16,43,45,67)(17,38,34,39)(19,46,48,24)(21,50,52,29)(25,54,55,51)(30,58,31,61)(37,63,65,41)(44,66,60,62)(53,57,68,64);(5,8,12,26,13,27,11,10)(6,15,34,46,18,42,17,24)(7,20,35,56,23,49,22,33)(9,25,31,50,32,55,30,29)(14,36,39,48,40,47,38,19)(16,41,60,68,45,63,44,53)(21,28,54,61,52,59,51,58)(37,62,64,67,65,66,57,43);(5,9,8,25,12,31,26,50,13,32,27,55,11,30,10,29)(6,16,24,53,17,44,42,63,18,45,46,68,34,60,15,41)(7,21,33,58,22,51,49,59,23,52,56,61,35,54,20,28)(14,37,36,62,39,64,48,67,40,65,47,66,38,57,19,43)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=fi,d^2=eh,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=eh,g^b=gh,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405769);