#365069

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1712 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)(7,15)(8,19,23,12)(9,17)(10,18)(11,14)(13,24)(16,22,27,21)(20,33,36,30)(25,31)(26,34,32,35)(28,29);(1,2,4,3)(5,7,15,6,11,18,10,14)(8,12,21,16,23,19,22,27)(9,17,29,13,25,31,24,28)(20,26,34,30,36,32,35,33);(5,8)(6,12)(7,16)(9,20)(10,21)(11,23)(13,26)(14,19)(15,22)(17,30)(18,27)(24,34)(25,36)(28,32)(29,35)(31,33);(1,3,4,2)(5,9,15,29,11,25,10,24)(6,13,18,31,14,28,7,17)(8,20,22,35,23,36,21,34)(12,26,27,33,19,32,16,30)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272265);