#7154812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
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 ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2321 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)(7,11)(10,12)(13,14)(15,21)(16,25)(17,19)(18,24)(20,26)(22,27)(23,28);(6,9)(7,11)(13,15,17,22)(14,18,19,28)(16,23,26,24)(20,27,25,21);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,19)(15,22)(20,25)(23,24);(6,9)(10,12)(15,23)(16,26)(18,27)(20,25)(21,28)(22,24);(5,7,8,11)(6,10,9,12)(15,22)(18,28)(21,27)(23,24)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307770, #72308123, #72308294
maximal normal subgroup of	#72307770, #72308123, #72308294

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7062008);