#392336

order	512
nilpotency class	3
isoclinism	isoclinic class of #184786
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1324 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1324 conjugacy classes of subgroups — too many to draw. Only its 314 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,c^b=ci,h^d=hi

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, 299532);