#493082

order	512
nilpotency class	4
isoclinism	isoclinic class of #33469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#442
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1762
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1115 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)(3,15,18,46)(4,10)(5,12)(6,29)(7,32)(8,14)(9,17,34,51)(11,27)(13,31)(16,45,47,62)(19,54,52,36)(20,56,53,35)(21,57,58,63)(22,38)(23,40)(24,42)(25,60)(26,43)(28,44)(30,39)(33,50,61,48)(37,55,64,49)(41,59);(1,3,6,19,31,53,28,51,8,18,27,52,7,20,5,17)(2,9,12,35,32,54,11,15,14,34,44,56,13,36,29,46)(4,21,24,48,59,16,26,49,22,58,30,50,25,47,23,55)(10,37,40,45,60,33,39,57,38,64,43,62,41,61,42,63);(1,4)(2,10)(3,16,18,47)(5,26)(6,30)(7,25)(8,22)(9,33,34,61)(11,42)(12,43)(13,41)(14,38)(15,45,46,62)(17,50,51,48)(19,55,52,49)(20,21,53,58)(23,28)(24,27)(29,39)(31,59)(32,60)(35,63,56,57)(36,37,54,64)(40,44);(4,22)(10,38)(16,47)(21,58)(23,26)(24,30)(25,59)(33,61)(37,64)(39,42)(40,43)(41,60)(45,62)(48,50)(49,55)(57,63)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285725, #72285899, #72286096, …
maximal normal subgroup of	#72285725, #72285899, #72286096, …
maximal quotient of	#72310171, #72342561

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400278);