#388371

order	512
nilpotency class	3
isoclinism	isoclinic class of #219706
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1268 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)(7,12,18,8)(9,22)(10,16)(11,15)(13,21,19,29)(14,27)(17,34,23,36)(20,35,28,30)(24,32)(25,31)(26,33);(5,7)(6,12)(8,16)(9,23)(10,18)(11,13)(14,28)(15,21)(17,24)(19,26)(20,31)(22,36)(25,30)(27,35)(29,33)(32,34);(2,4)(5,8,10,12)(6,13)(7,15)(9,20,24,28)(11,21,26,29)(14,30)(16,19)(17,32)(18,33)(22,23)(25,36,27,34)(31,35);(2,4)(5,9,10,24)(6,14,16,31)(7,17,18,23)(8,20,12,28)(11,27,26,25)(13,30,19,35)(15,32,33,22)(21,34,29,36)

Polycyclic presentation:

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

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, 295567);