#6496581

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1699 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)(23,31)(24,28)(25,37)(26,29)(27,34)(30,39)(32,44)(33,45)(35,42)(36,49)(38,40)(41,47)(43,48)(46,52)(50,51);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(23,32)(24,35)(27,40)(28,42)(31,44)(34,38)(36,48)(43,49);(7,13)(10,17)(14,19)(18,20)(21,23,26,32)(22,27,29,40)(24,33,35,46)(25,38,39,34)(28,41,42,51)(30,44,37,31)(36,50,48,47)(43,52,49,45);(7,13)(10,17)(14,19)(18,20)(21,24)(22,28)(23,33)(25,36)(26,35)(27,41)(29,42)(30,43)(31,45)(32,46)(34,47)(37,49)(38,50)(39,48)(40,51)(44,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,29)(24,35)(27,40)(30,37)(31,44)(33,46)(36,48)(47,50)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,e^2=hi,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, 6403777);