#391251

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

889 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298447);