#125159

order	512
nilpotency class	4
isoclinism	isoclinic class of #125092
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
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×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

383 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,7,8,5)(4,6)(9,10,13,19,14,20,34,49,15,21,33,50,28,18,46,47)(11,22,24,52,25,16,36,38,26,39,56,48,35,40,32,61)(12,17,29,43,30,44,60,70,31,45,59,71,53,42,67,68)(23,51,54,72,27,37,57,63,55,41,65,66,58,64,62,69);(1,3)(2,5)(4,8)(6,7)(9,11,15,26)(10,16,21,40)(12,27,31,58)(13,32,33,36)(14,35,28,25)(17,41,45,51)(18,39,20,22)(19,48,50,52)(23,53,55,30)(24,34,56,46)(29,54,59,65)(37,42,64,44)(38,47,61,49)(43,69,71,63)(57,60,62,67)(66,68,72,70);(9,12,28,53,15,31,14,30)(10,17,18,42,21,45,20,44)(11,23,35,58,26,55,25,27)(13,29,46,67,33,59,34,60)(16,37,22,51,40,64,39,41)(19,43,47,68,50,71,49,70)(24,54,32,62,56,65,36,57)(38,63,52,72,61,69,48,66)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=hi,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32355);