#97092

order	512
nilpotency class	3
isoclinism	isoclinic class of #97080
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

1070 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,7,8,5)(4,6)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22)(25,26,28,32,29,31,37,39)(27,33,35,40,34,30,36,38);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,29,34)(26,30,31,33)(28,36,37,35)(32,40,39,38);(1,3)(2,5)(4,8)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(27,34)(30,33)(35,36)(38,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,f^b=fg,f^c=fh

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, 4288);