#6434678

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432833
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341874);