#124975

order	512
nilpotency class	4
isoclinism	isoclinic class of #124937
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1533
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

384 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 90 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)(9,10,12,18,13,19,26,35,14,16,25,31,28,17,32,33)(11,20,22,34,23,29,38,39,21,36,37,40,27,15,24,30);(1,3)(2,5)(4,8)(6,7)(9,11,14,21)(10,15)(12,24,25,38)(13,27,28,23)(16,29)(17,20)(18,34)(19,36)(22,26,37,32)(30,35)(31,40)(33,39);(2,6)(5,7)(10,16)(12,25)(15,29)(17,19)(20,36)(22,37)(24,38)(26,32)

Polycyclic presentation:

2^9|a^2=e,b^2=g,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cfgh,d^a=df,g^a=gi,d^b=dg,e^b=efg,f^b=fi,h^b=hi,e^c=ei,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, 32171);