#392842

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×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1468 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)(5,6,11,18,10,16,12,17)(7,19,23,31,22,13,24,32)(8,21,27,33,25,36,26,14)(9,20,29,15,28,35,30,34);(1,2,4,3)(5,7,12,24,10,22,11,23)(6,13,17,31,16,19,18,32)(8,20,26,34,25,35,27,15)(9,21,30,14,28,36,29,33);(1,3,4,2)(5,8,10,25)(6,14,16,33)(7,20,22,35)(9,13,28,19)(11,27,12,26)(15,24,34,23)(17,36,18,21)(29,32,30,31);(5,9)(6,15)(7,21)(8,13)(10,28)(11,29)(12,30)(14,24)(16,34)(17,20)(18,35)(19,25)(22,36)(23,33)(26,31)(27,32)

Polycyclic presentation:

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

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, 300038);