#322615

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

726 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)(5,6,10,12,9,14,8,13)(7,11,17,18,15,19,16,20)(21,22,26,34,25,30,24,29)(23,27,31,28,33,36,32,35);(1,2,4,3)(5,7)(6,11)(8,16)(9,15)(10,17)(12,18)(13,20)(14,19)(21,23,26,31,25,33,24,32)(22,27,34,28,30,36,29,35);(6,12)(8,10)(11,18)(13,14)(16,17)(19,20)(21,23,25,33)(22,28,30,35)(24,31,26,32)(27,29,36,34);(1,3,4,2)(7,15)(11,19)(16,17)(18,20)(21,24,25,26)(22,29,30,34)(23,31,33,32)(27,28,36,35)

Polycyclic presentation:

2^9|a^2=fh,b^2=egi,c^2=i,d^2=ei,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, 229811);