#392802

order	512
nilpotency class	3
isoclinism	isoclinic class of #32888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299998);