#845161

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1820 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dfg,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 752357);