#348231

order	512
nilpotency class	3
isoclinism	isoclinic class of #346398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#447	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1120 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh,f^c=fh,g^c=gi,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, 255427);