#249122

order	512
nilpotency class	3
isoclinism	isoclinic class of #249079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1161 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156318);