#497411

order	512
nilpotency class	4
isoclinism	isoclinic class of #497408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3290	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1041 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=i,f^2=h,g^2=i,h^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=fhi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285481, #72285570, #72285693
maximal normal subgroup of	#72285481, #72285570, #72285693
maximal quotient of	#72287269, #72287484, #72287682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404607);