#6396715

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=fgh,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6303911);