#805036

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2080 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=fghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 712232);