#395126

order	512
nilpotency class	3
isoclinism	isoclinic class of #394102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60425	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1773 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fi,h^c=hi,f^d=fi,h^d=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, 302322);