#385545

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	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=ef,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, 292741);