#395949

order	512
nilpotency class	3
isoclinism	isoclinic class of #394202
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1327 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^c=fi,h^c=hi,f^d=fi

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, 303145);