#384628

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=dei,d^c=di

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, 291824);