#411854

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

2018 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)(7,19)(8,23)(9,22)(10,18)(11,16)(12,17)(13,21)(14,26)(15,28)(20,30)(24,36)(25,35)(27,33)(29,32)(31,34);(1,2,3,4)(5,7,10,21,12,22,11,15)(6,13,16,19,17,28,18,9)(8,24,26,35,27,20,29,31)(14,30,23,25,32,36,33,34);(5,8)(6,14)(7,20)(9,25)(10,29)(11,26)(12,27)(13,30)(15,31)(16,33)(17,32)(18,23)(19,34)(21,35)(22,24)(28,36);(5,9,11,28,12,19,10,13)(6,15,18,22,17,21,16,7)(8,25,26,36,27,34,29,30)(14,31,23,24,32,35,33,20)

Polycyclic presentation:

2^9|b^2=e,d^2=h,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, 319050);