#348252

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×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1124 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)(5,6,8,12)(7,14,15,10)(9,11,17,13)(16,19,20,18)(21,22)(23,27)(24,28)(25,26)(29,35)(30,32)(31,33)(34,36);(1,2,3,4)(5,7)(6,10)(8,15)(9,16)(11,18)(12,14)(13,19)(17,20)(22,26)(27,32)(28,33)(34,35);(6,11)(9,17)(10,18)(12,13)(14,19)(16,20)(21,23)(22,27)(24,29)(25,30)(26,32)(28,34)(31,36)(33,35);(7,15)(10,14)(16,20)(18,19)(21,24,25,31)(22,28,26,33)(23,29,30,36)(27,34,32,35)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,d^2=i,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, 255448);