#497342

order	512
nilpotency class	4
isoclinism	isoclinic class of #497339
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#61083	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#578 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1095 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)(4,10)(5,23)(6,12,28,9)(7,13)(8,17)(11,27)(15,39)(16,43)(18,42,45,58)(19,49)(20,40)(21,37,52,33)(22,38)(24,31)(25,36)(26,34,55,41)(29,53,32,48)(30,47)(35,44)(46,59,63,62)(50,57,64,56)(51,60,54,61);(1,3,7,15)(2,8,13,31)(4,19,22,47)(5,24,27,17)(6,18,28,45)(9,32,12,29)(10,35,38,43)(11,39,23,14)(16,25,44,20)(21,50,52,64)(26,53,55,48)(30,40,49,36)(33,59,37,62)(34,42,41,58)(46,54,63,51)(56,60,57,61);(2,9)(3,15)(6,28)(8,29)(10,33)(11,34)(12,13)(14,42)(16,44)(17,24)(19,47)(21,52)(23,41)(26,55)(30,56)(31,32)(35,62)(36,60)(37,38)(39,58)(40,61)(43,59)(49,57)(51,54);(1,4,7,22)(2,10,13,38)(3,16,15,44)(5,25,27,20)(6,21,28,52)(8,30,31,49)(9,33,12,37)(11,40,23,36)(14,43,39,35)(17,47,24,19)(18,46,45,63)(26,54,55,51)(29,56,32,57)(34,61,41,60)(42,59,58,62)(48,64,53,50)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=de,e^b=ei,f^c=fi,g^c=ghi,e^d=ei,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404538);