#147641

order	512
nilpotency class	4
isoclinism	isoclinic class of #137661
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,g^b=gi,h^b=hi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281756, #72281903, #72282164
maximal normal subgroup of	#72281756, #72281903, #72282164

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54837);