#379834

order	512
nilpotency class	3
isoclinism	isoclinic class of #29902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3010	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2072 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287030);