#7613477

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563341
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,22)(13,18)(14,17)(15,19)(20,23)(21,24);(2,5)(6,8)(9,11,15,21)(10,16,19,24)(12,13,20,14)(17,22,18,23);(1,3,4,7)(2,6,5,8)(9,12,14,11,15,20,13,21)(10,16,18,22,19,24,17,23);(10,17,19,18)(16,23,24,22);(2,5)(6,8)(9,13,15,14)(10,18,19,17)(11,20,21,12)(16,23,24,22)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217788
maximal normal subgroup of	#72217788
maximal quotient of	#72216143, #72216148, #72216158

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7520673);