#499888

order	512
nilpotency class	4
isoclinism	isoclinic class of #499885
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23900
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1843 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)(4,9)(5,10)(6,27)(7,13)(8,16)(11,25)(12,30)(15,46)(17,40)(18,51)(19,53)(20,37)(21,38)(22,43)(23,33)(24,35)(26,41)(28,45)(29,36)(31,49)(32,57)(34,54)(39,59)(42,50)(44,47)(48,60)(52,62)(55,61)(56,63)(58,64);(1,3,7,17)(2,8,13,33)(4,19,20,52)(5,23,25,16)(6,28,30,18)(9,34,37,61)(10,14,11,40)(12,42,27,32)(15,29,47,22)(21,55,24,54)(26,50,59,57)(31,43,60,36)(35,62,38,53)(39,51,41,45)(44,63,46,64)(48,56,49,58);(1,4)(2,9)(3,15)(5,24)(6,29)(7,20)(8,31)(10,38)(11,35)(12,43)(13,37)(14,44)(16,49)(17,47)(18,52)(19,28)(21,25)(22,30)(23,48)(26,56)(27,36)(32,61)(33,60)(34,42)(39,63)(40,46)(41,64)(45,62)(50,54)(51,53)(55,57)(58,59);(2,10)(4,20)(5,25)(8,14)(9,35)(11,13)(12,39)(15,47)(16,23)(19,52)(22,29)(26,59)(27,41)(31,46)(32,45)(33,40)(34,62)(36,63)(37,38)(42,51)(43,64)(44,60)(50,57)(53,61)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312282, #72312350, #72312372, …
maximal normal subgroup of	#72312282, #72312350, #72312372, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407084);