#395788

order	512
nilpotency class	3
isoclinism	isoclinic class of #394590
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1741 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,g^c=gi,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, 302984);