#388483

order	512
nilpotency class	3
isoclinism	isoclinic class of #30871
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
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

1179 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)(8,24)(9,15)(10,34)(11,16)(12,17)(13,21)(14,28)(19,48)(20,51)(22,38)(23,37)(25,47)(26,57)(27,42)(29,41)(30,62)(31,45)(32,44)(33,46)(35,50)(36,52)(39,54)(40,59)(43,53)(49,67)(55,60)(56,66)(58,63)(61,65)(64,68);(5,7)(6,13)(8,25)(9,30)(10,22)(11,21)(12,23)(14,39)(15,43)(16,37)(17,38)(18,34)(19,27)(20,31)(24,35)(26,58)(28,50)(29,54)(32,53)(33,52)(36,44)(40,63)(41,48)(42,47)(45,62)(46,51)(49,60)(55,66)(56,59)(57,64)(61,68)(65,67);(1,3,4,2)(5,8)(6,14)(7,19)(9,26)(10,27)(11,28)(12,29)(13,35)(15,40)(16,41)(17,42)(18,47)(20,49)(21,25)(22,54)(23,50)(24,34)(30,56)(31,59)(32,61)(33,60)(36,63)(37,39)(38,48)(43,66)(44,57)(45,55)(46,65)(51,64)(52,68)(53,58)(62,67);(5,9,10,31,12,33,11,32)(6,15,16,44,17,46,34,45)(7,20,21,30,23,53,22,52)(8,26,27,59,29,60,28,61)(13,36,18,43,38,62,37,51)(14,40,41,57,42,65,24,55)(19,49,25,56,50,58,54,68)(35,63,47,66,48,67,39,64)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,f^d=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, 295679);