#2241003

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1610 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fh,c^2=fi,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2148199);