#429605

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1057 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:

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

Polycyclic presentation:

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

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, 336801);