#497347

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	#61090	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#576 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1101 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=i,f^2=i,g^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404543);