#497336

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	#61088	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#577 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1113 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)(3,13)(4,19,22,36)(5,23)(6,26,28,11)(7,12)(8,16)(9,21)(10,24)(14,39,43,63)(15,32)(17,46,48,38)(18,30)(20,51,52,50)(25,35,56,55)(27,34)(29,59)(31,45,58,44)(33,57)(37,42)(40,62,41,49)(47,60)(53,54)(61,64);(1,3,7,18)(2,8,12,32)(4,14,22,43)(5,15,24,16)(6,17,28,48)(9,29,34,61)(10,30,23,13)(11,31,26,58)(19,49,36,62)(20,40,52,41)(21,42,27,47)(25,44,56,45)(33,60,54,37)(35,46,55,38)(39,50,63,51)(53,64,57,59);(1,4,7,22)(2,9,12,34)(3,14,18,43)(5,20,24,52)(6,27,28,21)(8,29,32,61)(10,33,23,54)(11,36,26,19)(13,37,30,60)(15,40,16,41)(17,47,48,42)(25,57,56,53)(31,62,58,49)(35,51,55,50)(38,63,46,39)(44,59,45,64);(3,15)(5,24)(8,30)(10,23)(13,32)(14,40)(16,18)(17,44)(20,52)(25,56)(29,60)(31,46)(33,54)(35,55)(37,61)(38,58)(39,62)(41,43)(42,64)(45,48)(47,59)(49,63)(50,51)(53,57)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^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	#72312282, #72312369, #72312381, …
maximal normal subgroup of	#72312282, #72312369, #72312381, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404532);