#6497646

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6404842);