#411856

order	512
nilpotency class	3
isoclinism	isoclinic class of #397514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2066 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)(5,6,12,16)(7,11,15,10)(8,17,22,19)(9,18,25,29)(13,26,28,21)(14,23,30,24)(20,33,35,27)(31,36,32,34);(1,2,3,4)(5,7,10,6,12,15,11,16)(8,19,21,13,22,17,26,28)(9,18,23,14,25,29,24,30)(20,32,34,27,35,31,36,33);(5,8)(6,13)(7,17)(9,20)(10,26)(11,21)(12,22)(14,27)(15,19)(16,28)(18,31)(23,36)(24,34)(25,35)(29,32)(30,33);(5,9,10,23,12,25,11,24)(6,14,15,29,16,30,7,18)(8,20,26,36,22,35,21,34)(13,27,19,32,28,33,17,31)

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=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, 319052);