#698353

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1820 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=f,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 605549);