#392282

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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

1706 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)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,25,21,24,29,31)(19,26,27,32,20,23,28,30);(1,2,4,3)(10,13)(14,16)(17,19)(18,23)(20,21)(22,27)(24,26)(25,30)(28,29)(31,32);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,23,24,26)(22,28,29,27)(25,30,31,32);(6,8)(19,20)(23,26)(27,28)(30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dg,c^b=ci,d^b=di,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299478);