#246612

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
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	C₂×C₄²	C₂⁴
prime radical	C₂×#3207	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1283 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,16)(8,12)(9,13)(10,15)(11,17)(14,19)(18,20)(21,22)(23,31)(24,27)(25,30)(26,29)(28,32)(33,36)(34,35);(1,2)(3,4)(5,7,10,8)(6,11,15,20)(9,18,19,17)(12,13,16,14)(22,27)(24,25)(28,35)(29,30)(31,36)(32,33);(2,4)(5,8,10,7)(6,12,15,16)(9,18,19,17)(11,13,20,14)(21,23,26,34)(22,28,29,36)(24,32,25,33)(27,35,30,31);(6,13)(9,19)(11,16)(12,20)(14,15)(17,18)(21,24)(22,29)(23,32)(25,26)(28,36)(33,34)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,f^b=fi,e^c=ei,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153808);