#261426

order	512
nilpotency class	3
isoclinism	isoclinic class of #246079
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1036 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=hi,d^2=ehi,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,f^c=fh,g^c=gh

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, 168622);