#246186

order	512
nilpotency class	3
isoclinism	isoclinic class of #246159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

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

1227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153382);