#6591906

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256922, #72256940, #72256974, …
maximal normal subgroup of	#72256922, #72256940, #72256974, …
maximal quotient of	#72264576

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6499102);