#368866

order	512
nilpotency class	3
isoclinism	isoclinic class of #24408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1658 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)(7,17)(8,22)(9,27)(10,30)(11,16)(12,20)(13,26)(14,28)(15,33)(18,42)(19,35)(21,40)(23,52)(24,55)(25,38)(29,41)(31,36)(32,39)(34,45)(37,58)(43,46)(44,61)(47,63)(48,62)(49,54)(50,56)(51,59)(53,67)(57,60)(64,65)(66,68);(1,3,4,2)(5,7)(6,12)(8,23)(9,19)(10,31)(11,21)(13,34)(14,35)(15,40)(16,36)(17,30)(18,26)(20,33)(22,49)(24,53)(25,45)(27,43)(28,48)(29,47)(32,46)(37,61)(38,52)(39,63)(41,62)(42,51)(44,57)(50,67)(54,59)(55,68)(56,65)(58,66)(60,64);(5,8,11,26)(6,13)(7,18)(9,24,29,57)(10,25,33,59)(12,34,36,49)(14,37)(15,38)(16,22)(17,42,40,52)(19,44)(20,45)(21,23)(27,60)(28,56,32,58)(30,51)(31,54)(35,61,63,67)(39,50)(41,55)(43,64,62,68)(46,65)(47,53)(48,66);(5,9,11,29)(6,14,16,39)(7,19,21,47)(8,24,26,57)(10,32,33,28)(12,35,36,63)(13,37,22,50)(15,41,30,27)(17,43,40,62)(18,44,23,53)(20,48,31,46)(25,58,59,56)(34,61,49,67)(38,55,51,60)(42,64,52,68)(45,66,54,65)

Polycyclic presentation:

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

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, 276062);