#93525

order	512
nilpotency class	4
isoclinism	isoclinic class of #93523
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×C₄×Q₈	C₂²
prime radical	C₂⁴	#95
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄×Q₈ ▷ C₂⁴ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 481 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=i,e^2=i,b^a=bc,c^a=cf,e^a=efgh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,h^b=hi,e^d=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, 721);