#7006672

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

2191 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)(13,14,15,23,18,22,25,19)(16,21,26,28,17,27,24,20);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,25)(14,19,22,23)(16,26,17,24)(20,27,28,21);(5,7)(6,9)(8,12)(10,11)(13,16)(14,20)(15,24)(17,18)(19,21)(22,28)(23,27)(25,26);(1,3,4,2)(7,12)(9,11)(13,17,18,16)(14,21,22,27)(15,24,25,26)(19,20,23,28);(6,10)(9,11)(13,15,18,25)(14,19,22,23)(16,24,17,26)(20,21,28,27)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,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, #72308547
maximal normal subgroup of	#72308532, #72308547

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6913868);