#94135

order	512
nilpotency class	5
isoclinism	isoclinic class of #7493
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1390
commutator	C₂×D₄	C₄×C₈
Frattini, cosocle	C₄×#124	C₂²
prime radical	C₂²×D₄	C₄²
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ C₄×#124 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

238 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=g,e^2=gi,f^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=dfi,f^b=fi,h^b=hi,d^c=dhi,f^c=fi,f^d=fi

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, 1331);