#152995

order	512
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

316 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=ei,d^2=gh,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,f^b=fg,g^b=gh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60191);