#249042

order	512
nilpotency class	3
isoclinism	isoclinic class of #249019
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	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 156238);