#394045

order	512
nilpotency class	3
isoclinism	isoclinic class of #394044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,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, 301241);