#130002

order	512
nilpotency class	4
isoclinism	isoclinic class of #129769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#135	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 902 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) = 72
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 37198);