#93295

order	512
nilpotency class	4
isoclinism	isoclinic class of #93295
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1276
commutator	C₄²	C₄×C₈
Frattini, cosocle	C₄×#117	C₂²
socle	C₂	#7017
prime radical	C₂⁴	#99
LCS	G ▷ C₄² ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #376 ◁ C₄×#117 ◁ G
LECS	G ▷ C₄×#117 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

319 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,d^2=h,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,c^b=ci,d^b=dg,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 491);