#674135

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2246 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)(3,8)(4,7)(5,6)(10,12)(14,15)(17,18,22,21)(19,20,24,23)(25,26,30,29)(27,32,31,28);(2,6)(7,8)(9,10,13,12)(11,15,16,14)(18,21)(19,24);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(17,19,22,24)(18,23,21,20)(25,27)(26,28)(29,32)(30,31);(1,4,5,3)(2,8,6,7)(9,12,13,10)(11,15,16,14)(17,20)(18,19)(21,24)(22,23)(25,28,30,32)(26,31,29,27);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,15)(13,16)(17,21,22,18)(19,20,24,23)(25,29,30,26)(27,32,31,28)

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=fgi,d^2=gh,e^2=fgh,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, 581331);