#524716

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3710 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgi,d^2=fghi,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 431912);