#512310

order	512
nilpotency class	4
isoclinism	isoclinic class of #511896
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=efi,d^2=eh,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=gh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419506);