#608780

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=f,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 515976);