#93127

order	512
nilpotency class	4
isoclinism	isoclinic class of #93119
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×#126	C₂²
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×#126 ▷ C₂⁴ ▷ C₂² ▷ C₁

332 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2)(3,14,16,37)(4,19,21,34)(5,24,25,10)(6,13)(7,12)(8,15,28,41)(9,23,32,52)(11,26)(17,46)(18,39,45,48)(20,30,43,31)(22,53,54,56)(27,35,57,36)(29,42,59,62)(33,40,49,51)(38,44)(47,64)(50,58,61,55)(60,63);(1,3,5,17,26,45,57,63)(2,8,10,29,11,28,35,58)(4,20,22,50,52,43,54,62)(6,16,25,44,7,18,27,47)(9,14,33,38,19,48,49,64)(12,30,36,59,13,31,24,55)(15,40,42,21,41,51,61,23)(32,39,53,60,34,37,56,46)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,e^2=hi,b^a=bc,c^a=cf,e^a=egi,f^a=fh,g^a=gi,c^b=cg,d^b=dfi,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323);