#500648

order	512
nilpotency class	4
isoclinism	isoclinic class of #500646
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24196
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33595	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

756 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)(3,8)(4,18)(5,24)(6,30)(7,13)(9,22)(10,28)(11,44)(12,27)(14,49)(15,42)(16,48)(17,37)(19,33)(20,26)(21,40)(23,46)(25,34)(29,47)(31,36)(32,43)(35,62)(38,52)(39,50)(41,58)(45,55)(51,54)(53,59)(56,64)(57,60)(61,63);(1,3,7,17)(2,8,13,37)(4,19,21,60)(5,25,29,62)(6,31,27,16)(9,38,39,49)(10,42,11,45)(12,48,30,36)(14,50,52,22)(15,44,55,28)(18,57,40,33)(20,59,41,64)(23,61,32,54)(24,34,47,35)(26,56,58,53)(43,63,46,51);(1,4)(2,9)(3,14)(5,26)(6,32)(7,21)(8,33)(10,43)(11,46)(12,20)(13,39)(15,54)(16,56)(17,52)(18,24)(19,25)(22,28)(23,27)(29,58)(30,41)(31,53)(34,64)(35,59)(36,51)(37,57)(38,45)(40,47)(42,49)(44,50)(48,63)(55,61)(60,62);(1,2)(3,8)(4,20,21,41)(5,10)(6,12)(7,13)(9,32,39,23)(11,29)(14,51,52,63)(15,34)(16,36)(17,37)(18,58,40,26)(19,59,60,64)(22,46,50,43)(24,28)(25,42)(27,30)(31,48)(33,56,57,53)(35,55)(38,54,49,61)(44,47)(45,62)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307951
maximal normal subgroup of	#72307951

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407844);