#391203

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1098 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298399);