#794199

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fgi,c^2=gh,d^2=fh,e^2=fg,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, 701395);