#375761

order	512
nilpotency class	3
isoclinism	isoclinic class of #23892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,f^d=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, 282957);