#379828

order	512
nilpotency class	3
isoclinism	isoclinic class of #379827
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14255
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2053 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)(3,13,17,8)(4,9)(5,10)(6,12)(7,11)(14,40,43,27)(15,41,44,28)(16,38,24,29)(18,47,46,30)(19,31)(20,33)(21,32)(22,35)(23,36)(25,39)(26,37)(34,49)(42,55,63,54)(45,62,52,56)(48,59,64,57)(50,58)(51,61)(53,60);(1,3)(2,8)(4,18)(5,15)(6,24)(7,17)(9,30)(10,28)(11,13)(12,38)(14,20)(16,26)(19,48)(21,46)(22,44)(23,52)(25,43)(27,33)(29,37)(31,57)(32,47)(34,59)(35,41)(36,62)(39,40)(42,50)(45,51)(49,64)(53,63)(54,58)(55,60)(56,61);(1,4,7,21)(2,9,11,32)(3,14,17,43)(5,19,22,49)(6,25,26,20)(8,27,13,40)(10,34,35,31)(12,39,37,33)(15,42,44,63)(16,46,24,18)(23,53,51,50)(28,55,41,54)(29,47,38,30)(36,58,61,60)(45,64,52,48)(56,57,62,59);(1,5)(2,10)(3,15)(4,19)(6,23)(7,22)(8,28)(9,31)(11,35)(12,36)(13,41)(14,42)(16,45)(17,44)(18,48)(20,50)(21,49)(24,52)(25,53)(26,51)(27,54)(29,56)(30,57)(32,34)(33,58)(37,61)(38,62)(39,60)(40,55)(43,63)(46,64)(47,59)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286082, #72286090, #72286253
maximal normal subgroup of	#72286082, #72286090, #72286253

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287024);