#129376

order	512
nilpotency class	4
isoclinism	isoclinic class of #12725
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2273 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

924 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,f^b=fg,e^c=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36572);