#318360

order	512
nilpotency class	3
isoclinism	isoclinic class of #262851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eghi,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225556);