#115634

order	512
nilpotency class	3
isoclinism	isoclinic class of #115631
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1729 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1729 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,7,8,5)(4,6)(9,10,13,18,14,17,26,32)(11,19,21,33,22,15,25,28)(12,16,24,31,23,30,36,40)(20,29,35,39,34,27,37,38);(9,11,14,22)(10,15,17,19)(12,20,23,34)(13,25,26,21)(16,29,30,27)(18,33,32,28)(24,37,36,35)(31,38,40,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,20)(13,24)(14,23)(15,27)(17,30)(18,31)(19,29)(21,35)(22,34)(25,37)(26,36)(28,38)(32,40)(33,39)

Polycyclic presentation:

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

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, 22830);