#457089

order	512
nilpotency class	3
isoclinism	isoclinic class of #456417
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1113 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)(5,6,9,14,11,15,10,25)(7,12,17,27,18,28,19,35)(8,20,22,31,23,30,24,13)(16,32,33,29,21,36,34,26);(1,2,4,3)(5,7,9,17,11,18,10,19)(6,12,14,27,15,28,25,35)(8,21,22,34,23,16,24,33)(13,29,20,36,31,26,30,32);(1,2)(3,4)(5,8)(6,13)(7,16)(9,24)(10,22)(11,23)(12,26)(14,30)(15,31)(17,34)(18,21)(19,33)(20,25)(27,36)(28,29)(32,35);(1,3,4,2)(5,6,11,15)(7,12,18,28)(8,13,23,31)(9,14,10,25)(16,26,21,29)(17,27,19,35)(20,24,30,22)(32,34,36,33)

Polycyclic presentation:

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

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, 364285);