#950618

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2744 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)(9,10,13,14)(11,12,15,16)(18,19)(20,24)(26,29)(27,31);(2,3)(6,7)(10,14)(11,15)(17,18,21,19)(20,22,24,23)(25,26)(27,30)(28,29)(31,32);(2,3)(6,7)(10,14)(11,15)(18,19)(22,23)(27,31)(30,32);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,15)(14,16)(17,19,21,18)(20,23,24,22)(25,27,28,31)(26,30,29,32);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,20,21,24)(18,22,19,23)(25,26)(27,30)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,d^2=i,e^2=h,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, 857814);