#7064668

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2092 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=hi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308532, #72308641
maximal normal subgroup of	#72308532, #72308641

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6971864);