#495267

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1982
prime radical	#60643	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1247 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285581, #72285937, #72285959, …
maximal normal subgroup of	#72285581, #72285937, #72285959, …
maximal quotient of	#72310118, #72342528

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402463);