#664125

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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

1829 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=fi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 571321);