#521592

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3758 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=h,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, 428788);