#386077

order	512
nilpotency class	3
isoclinism	isoclinic class of #232128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1579 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:

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

Polycyclic presentation:

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

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, 293273);