#494480

order	512
nilpotency class	4
isoclinism	isoclinic class of #13005
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1874
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1874
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1347 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285693, #72285705, #72285805, …
maximal normal subgroup of	#72285693, #72285705, #72285805, …
maximal quotient of	#72321137, #72322782, #72342545

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401676);