#622418

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×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

3062 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fhi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 529614);