#361800

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1913 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)(7,12,9,14)(8,19,21,31)(10,16)(11,15)(13,22,29,25)(17,33)(18,27,23,30)(20,36)(24,32)(26,35)(28,34);(1,2,3,4)(5,7,10,9)(6,12,16,14)(8,17,21,20)(11,23,24,18)(13,28,29,26)(15,30,32,27)(19,33,31,36)(22,35,25,34);(2,4)(5,8)(6,13)(7,17)(9,20)(10,21)(11,25)(12,26)(14,28)(15,31)(16,29)(18,35)(19,32)(22,24)(23,34)(27,36)(30,33);(5,9,10,7)(6,14,16,12)(8,20,21,17)(11,23,24,18)(13,28,29,26)(15,30,32,27)(19,36,31,33)(22,35,25,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268996);