#497839

order	512
nilpotency class	4
isoclinism	isoclinic class of #497838
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

586 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)(3,17)(4,22,26,55)(5,12)(6,15)(7,33)(8,13)(9,16)(10,20)(11,25,40,57)(14,34)(18,48,51,64)(19,46)(21,43)(23,53,59,47)(24,56,27,54)(28,45)(29,44)(30,36)(31,37)(32,42)(35,58,63,61)(38,50,52,60)(39,62,41,49);(1,3,9,21)(2,10,16,37)(4,18,26,51)(5,28,30,7)(6,31,8,20)(11,35,40,63)(12,14,36,46)(13,43,15,17)(19,32,34,29)(22,52,55,38)(23,57,59,25)(24,60,27,50)(33,44,45,42)(39,53,41,47)(48,54,64,56)(49,61,62,58);(1,4,7,25,8,27,34,62,9,26,28,57,6,24,19,49)(2,11,14,22,15,41,33,56,16,40,46,55,13,39,45,54)(3,18,5,23,20,50,29,58,21,51,30,59,31,60,32,61)(10,35,36,52,17,47,44,48,37,63,12,38,43,53,42,64);(1,5)(2,12)(3,19)(4,23)(6,32)(7,20)(8,29)(9,30)(10,33)(11,38)(13,44)(14,43)(15,42)(16,36)(17,46)(18,49)(21,34)(22,53)(24,61)(25,50)(26,59)(27,58)(28,31)(35,56)(37,45)(39,48)(40,52)(41,64)(47,55)(51,62)(54,63)(57,60)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285521, #72285570, #72285601
maximal normal subgroup of	#72285521, #72285570, #72285601
maximal quotient of	#72287405, #72287564, #72309554, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405035);