#6467665

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,21)(18,24)(22,27)(23,26)(25,28);(1,2,3,4)(15,22)(17,23)(18,25)(19,27)(21,26)(24,28);(6,9)(10,12)(13,15,23,28,16,24,17,22)(14,18,26,27,20,19,21,25);(13,15,23,28,16,24,17,22)(14,19,26,25,20,18,21,27);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|b^2=f,c^2=hi,d^2=hi,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286953, #72286976, #72286991
maximal normal subgroup of	#72286953, #72286976, #72286991
maximal quotient of	#72287773

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374861);