#129562

order	512
nilpotency class	4
isoclinism	isoclinic class of #129559
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

592 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ch,d^b=dg,g^b=gi,f^c=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, 36758);