#7003538

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
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
prime radical	C₂×#120	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1944 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6910734);