#131397

order	512
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

614 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dgi,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38593);