#374121

order	512
nilpotency class	3
isoclinism	isoclinic class of #374120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14134
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

2364 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)(3,14)(4,20,24,35)(5,25)(6,28)(7,12)(8,13)(9,17)(10,23,36,30)(11,26)(15,31,21,49)(16,32)(18,33)(19,39)(22,51,54,34)(27,37)(29,38)(40,43,56,47)(41,44)(42,61,50,60)(45,58)(46,57)(48,53,59,52)(55,63,64,62);(1,3,8,19)(2,9,13,33)(4,21,24,15)(5,16,27,45)(6,18,7,17)(10,31,36,49)(11,32,38,58)(12,39,28,14)(20,40,35,56)(22,50,54,42)(23,43,30,47)(25,41,37,57)(26,46,29,44)(34,55,51,64)(48,60,59,61)(52,62,53,63);(1,4,7,30,8,24,6,23)(2,10,12,35,13,36,28,20)(3,15,18,47,19,21,17,43)(5,22,26,52,27,54,29,53)(9,31,14,40,33,49,39,56)(11,34,37,59,38,51,25,48)(16,42,44,62,45,50,46,63)(32,55,41,60,58,64,57,61);(1,5)(2,11)(3,16)(4,22)(6,29)(7,26)(8,27)(9,32)(10,34)(12,37)(13,38)(14,41)(15,42)(17,46)(18,44)(19,45)(20,48)(21,50)(23,53)(24,54)(25,28)(30,52)(31,55)(33,58)(35,59)(36,51)(39,57)(40,60)(43,63)(47,62)(49,64)(56,61)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,e^2=i,f^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,h^c=hi,e^d=ei

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306448, #72306455, #72306472
maximal normal subgroup of	#72306448, #72306455, #72306472

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281317);