#298253

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
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
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1002 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)(7,16)(8,19)(9,15)(10,14)(11,13)(12,18)(17,20)(21,22)(23,29)(24,27)(25,34)(26,31)(28,33)(30,35)(32,36);(1,2,4,3)(5,7,10,18)(6,12,14,16)(8,11,17,9)(13,19,15,20)(21,23)(22,26)(24,31)(25,35)(27,29)(28,30)(32,33)(34,36);(5,8,11,18,10,17,9,7)(6,12,15,19,14,16,13,20)(21,22)(23,30,31,36)(24,27)(25,28)(26,35,29,32)(33,34);(1,3,4,2)(5,9,10,11)(6,13,14,15)(7,17,18,8)(12,20,16,19)(22,27)(26,29)(28,34)(30,36)

Polycyclic presentation:

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

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, 205449);