#497337

order	512
nilpotency class	4
isoclinism	isoclinic class of #497333
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	#61091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#578 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1089 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)(8,16)(11,27)(15,43)(17,44,49,42)(18,39)(19,50)(20,40)(21,37,53,33)(22,38)(24,32)(25,36)(26,34,56,41)(29,48,31,54)(30,47)(35,46)(45,59,63,62)(51,57,64,58)(52,60,55,61);(1,3,7,18)(2,8,13,32)(4,19,22,47)(5,24,27,16)(6,17,28,49)(9,29,12,31)(10,35,38,43)(11,39,23,14)(15,25,46,20)(21,51,53,64)(26,54,56,48)(30,40,50,36)(33,59,37,62)(34,44,41,42)(45,55,63,52)(57,61,58,60);(2,9)(6,28)(8,29)(10,33)(11,34)(12,13)(14,42)(17,49)(21,53)(23,41)(26,56)(30,57)(31,32)(35,59)(36,60)(37,38)(39,44)(40,61)(43,62)(45,63)(48,54)(50,58)(51,64)(52,55);(1,4,7,22)(2,10,13,38)(3,15,18,46)(5,25,27,20)(6,21,28,53)(8,30,32,50)(9,33,12,37)(11,40,23,36)(14,43,39,35)(16,47,24,19)(17,45,49,63)(26,55,56,52)(29,57,31,58)(34,61,41,60)(42,62,44,59)(48,64,54,51)

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,d^b=de,e^b=ei,f^c=fi,g^c=ghi,e^d=ei,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342045
maximal normal subgroup of	#72342045

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404533);