#500939

order	512
nilpotency class	4
isoclinism	isoclinic class of #500938
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24338
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

687 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)(3,14,17,46)(4,10)(5,24)(6,28)(7,32)(8,13)(9,18,35,15)(11,25)(12,27)(16,49,50,45)(19,42,33,47)(20,52,54,63)(21,55)(22,57)(23,41)(26,44,30,36)(29,43)(31,60)(34,48,62,51)(37,53,64,59)(38,56)(39,61)(40,58);(3,15,17,18)(5,25)(6,29)(7,27)(9,14,35,46)(11,24)(12,32)(16,48,50,51)(19,30,33,26)(20,53,54,59)(21,56)(22,58)(28,43)(31,61)(34,45,62,49)(36,42,44,47)(37,63,64,52)(38,55)(39,60)(40,57);(1,3)(2,9)(4,20)(5,26)(6,30)(7,33)(8,17)(10,37)(11,42)(12,44)(13,35)(14,28)(15,25)(16,22)(18,29)(19,27)(21,51)(23,54)(24,46)(31,48)(32,36)(34,40)(38,45)(39,49)(41,64)(43,47)(50,58)(52,60)(53,61)(55,63)(56,59)(57,62);(1,4,8,23)(2,10,13,41)(3,16)(5,21,6,31)(7,22,27,58)(9,34)(11,38,43,39)(12,40,32,57)(14,45)(15,48)(17,50)(18,51)(19,20)(24,55,28,60)(25,56,29,61)(26,59)(30,53)(33,54)(35,62)(36,37)(42,63)(44,64)(46,49)(47,52)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250066, #72307951
maximal normal subgroup of	#72250066, #72307951
maximal quotient of	#72250668, #72281594, #72282077

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408135);