#698647

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2232 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

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, 605843);