#436654

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1170 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)(7,18,23,34)(8,24,28,14)(9,27,30,26)(10,15)(11,17)(12,16)(13,21,33,22)(19,35)(20,32)(25,29)(31,36);(5,7)(6,13,16,33)(8,25)(9,29,30,31)(10,21)(11,22)(12,23)(14,35,24,20)(15,18,17,34)(19,26)(27,32)(28,36);(5,8)(6,14)(7,19)(9,17)(10,26)(11,27)(12,28)(13,31)(15,30)(16,24)(18,20)(21,36)(22,25)(23,32)(29,33)(34,35);(2,4)(5,9,10,24,12,30,11,14)(6,8,17,26,16,28,15,27)(7,20,21,31,23,35,22,29)(13,32,34,25,33,19,18,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343850);