#6389672

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	C₂×#121	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1722 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)(23,28)(24,36)(25,30)(26,40)(27,31)(29,39)(32,45)(33,42)(34,43)(35,49)(37,44)(38,50)(41,47)(46,51)(48,52);(1,3,4,2)(5,7)(6,11)(8,15)(9,17)(10,16)(12,19)(13,14)(18,20)(21,23)(22,28)(24,32)(25,33)(26,35)(27,34)(29,41)(30,42)(31,43)(36,45)(37,46)(38,48)(39,47)(40,49)(44,51)(50,52);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24,26,39,27,38,25,37)(22,29,30,36,31,44,40,50)(23,32,35,47,34,48,33,46)(28,41,42,45,43,51,49,52);(7,16)(11,14)(15,17)(19,20)(21,25,27,26)(22,30,31,40)(23,33,34,35)(24,37,38,39)(28,42,43,49)(29,36,44,50)(32,46,48,47)(41,45,51,52);(5,9,10,8)(6,12,13,18)(7,17,16,15)(11,19,14,20)(23,34)(24,38)(28,43)(29,44)(33,35)(36,50)(37,39)(42,49)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,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, 6296868);