#117634

order	512
nilpotency class	3
isoclinism	isoclinic class of #117631
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1386 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,e^c=eh,g^c=gh

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, 24830);