#395947

order	512
nilpotency class	3
isoclinism	isoclinic class of #394199
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

1351 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)(7,17)(8,19)(9,21,24,14)(10,25)(11,16)(12,13)(15,26)(18,20,35,33)(22,31,23,32)(27,29,36,34)(28,30);(1,2)(3,4)(5,7,10,13,11,19,26,30)(6,12,15,17,16,28,25,8)(9,18,22,29,24,35,23,34)(14,27,31,33,21,36,32,20);(2,4)(5,8,10,12,11,17,26,28)(6,13,15,19,16,30,25,7)(9,20,22,27,24,33,23,36)(14,29,31,35,21,34,32,18);(5,9)(6,14)(7,18)(8,20)(10,22)(11,24)(12,27)(13,29)(15,31)(16,21)(17,33)(19,35)(23,26)(25,32)(28,36)(30,34)

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,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, 303143);