#928140

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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

1892 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)(3,5)(4,7)(6,8)(11,15)(13,16)(17,18)(19,22)(20,24)(21,23)(25,26,28,30)(27,29,32,31);(2,3)(6,7)(17,19,21,20)(18,22,23,24)(25,27)(26,29)(28,32)(30,31);(2,3)(6,7)(9,10,12,14)(11,13,15,16)(17,20,21,19)(18,22,23,24)(25,26)(27,31)(28,30)(29,32);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,18)(19,24)(20,22)(21,23)(25,26)(27,31)(28,30)(29,32);(1,4)(2,6)(3,7)(5,8)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=gi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 835336);