#298343

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205539);