#950031

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

1846 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)(4,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,23,24,20)(25,26)(27,28)(29,31)(30,32);(1,3,6,8)(2,7,4,5)(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,27,29,32)(26,28,31,30);(2,4)(3,8)(9,10,12,14)(11,13,16,15)(17,20,21,23)(18,19,22,24)(25,26,29,31)(27,30,32,28);(1,4,6,2)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,23)(22,24)(25,27)(26,30)(28,31)(29,32);(1,5,6,7)(2,8,4,3)(9,11)(10,13)(12,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,28,29,30)(26,27,31,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=fg,d^2=fhi,e^2=g,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, 857227);