#392357

order	512
nilpotency class	3
isoclinism	isoclinic class of #184786
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1566 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1566 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=e,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, 299553);