#7003493

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
prime radical	#59017	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1942 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)(14,19)(17,20)(21,27)(22,25)(23,28)(24,26);(5,6)(7,11)(8,10)(9,12)(13,14,20,28,18,23,17,19)(15,21,26,22,16,25,24,27);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,16)(14,21,23,25)(17,24,20,26)(19,27,28,22);(7,12)(9,11)(13,16,18,15)(14,22,23,27)(17,24,20,26)(19,25,28,21);(6,10)(9,11)(14,23)(19,28)(21,25)(22,27)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^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,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6910689);