#680927

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

1862 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)(3,7)(4,8)(5,6)(10,11)(14,15)(17,18,21,20)(19,23,24,22)(25,26,30,29)(27,28,31,32);(2,6)(4,7)(18,20)(22,23)(26,29)(27,31);(9,10,13,11)(12,14,16,15)(17,18,21,20)(19,22,24,23)(25,27)(26,28)(29,32)(30,31);(1,3,5,8)(2,7,6,4)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,23)(21,24)(25,28,30,32)(26,31,29,27);(1,4,5,7)(2,8,6,3)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,23,24,22)(25,29,30,26)(27,32,31,28)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=fi,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, 588123);