#393176

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1174 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300372);