#359891

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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

2091 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,f^2=i,h^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,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, 267087);