#818470

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,13)(14,15)(18,22)(19,24)(25,26,29,28)(27,30,32,31);(2,4)(6,7)(10,15)(13,16)(17,18)(19,20)(21,22)(23,24)(25,26,29,28)(27,31,32,30);(1,3)(2,6)(4,7)(5,8)(9,11,14,12)(10,16,15,13)(17,19)(18,23)(20,22)(21,24)(25,27,29,32)(26,30,28,31);(1,3)(2,7)(4,6)(5,8)(9,12,14,11)(10,16,15,13)(17,19)(18,20)(21,24)(22,23);(1,4,5,2)(3,7,8,6)(9,13,14,16)(10,12,15,11)(17,20)(18,24)(19,22)(21,23)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=fghi,c^2=hi,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 725666);