#464207

order	512
nilpotency class	3
isoclinism	isoclinic class of #454399
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60643	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1707 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,g^c=gi,f^d=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, 371403);