#1435453

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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

1911 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=f,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

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, 1342649);