#6381473

order	512
nilpotency class	3
isoclinism	isoclinic class of #61106
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3020	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3083 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,d^b=df,f^b=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288669);