#129518

order	512
nilpotency class	4
isoclinism	isoclinic class of #12741
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

797 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 797 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

2^9|a^2=f,b^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,d^b=dg,g^b=gi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36714);