#456583

order	512
nilpotency class	3
isoclinism	isoclinic class of #456417
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1379 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)(5,6,10,16,9,15,11,27)(7,17,21,32,20,12,22,29)(8,23,25,14,24,31,26,13)(18,33,34,28,19,35,36,30);(2,4)(5,7,11,22,9,20,10,21)(6,12,27,32,15,17,16,29)(8,19,26,34,24,18,25,36)(13,30,31,35,14,28,23,33);(1,2)(3,4)(5,8,9,24)(6,13,15,14)(7,18,20,19)(10,26,11,25)(12,28,17,30)(16,31,27,23)(21,36,22,34)(29,33,32,35);(2,4)(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,28)(13,15)(16,31)(17,30)(18,20)(21,36)(22,34)(23,27)(29,35)(32,33)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,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, 363779);