#93515

order	512
nilpotency class	4
isoclinism	isoclinic class of #93514
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×#117	C₂²
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×#117 ▷ C₂⁴ ▷ C₂ ▷ C₁

546 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 64

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 711);