#865721

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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

1703 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)(4,7)(10,13)(14,16)(17,18,22,20)(19,21,24,23)(26,28)(30,31);(1,2)(3,7)(4,8)(5,6)(10,13)(14,16)(18,20)(19,24)(25,26,29,28)(27,31,32,30);(9,10)(11,14)(12,13)(15,16)(17,19,22,24)(18,23,20,21)(25,27,29,32)(26,30,28,31);(1,3,5,8)(2,7,6,4)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,21,24,23)(25,26,29,28)(27,30,32,31);(1,4,5,7)(2,8,6,3)(9,11,12,15)(10,14,13,16)(17,21)(18,19)(20,24)(22,23)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 772917);