#6467637

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2403 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239652, #72256943, #72256973, …
maximal normal subgroup of	#72239652, #72256943, #72256973, …
maximal quotient of	#72237235, #72239613, #72264592

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374833);