#462213

order	512
nilpotency class	3
isoclinism	isoclinic class of #461251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

701 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=efi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369409);