#359575

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13867
prime radical	#63090	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3677 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317339, #72317399
maximal normal subgroup of	#72317339, #72317399

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266771);