#705151

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

1854 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fh,c^2=fi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,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, 612347);