#379830

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	#60368	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2026 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,g^2=i,h^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	#72306344, #72306448, #72306465
maximal normal subgroup of	#72306344, #72306448, #72306465

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287026);