#610653

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2686 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)(7,8)(10,13)(11,15)(17,18,20,22)(19,23,24,21)(25,26)(27,31)(28,30)(29,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,20,24)(18,21,22,23)(25,27)(26,29)(28,32)(30,31);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(18,22)(19,24)(26,30)(29,31);(2,6)(7,8)(10,13)(14,16)(19,24)(21,23)(26,30)(27,32);(1,4,5,3)(2,8,6,7)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 517849);