#6481453

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3493	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2926 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6388649);