#322580

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

734 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229776);