#459437

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14,20,25)(15,27,31,40)(16,33,36,23)(17,37,35,22)(18,26,19,24)(21,32,41,30)(28,38,34,44)(29,39,42,43);(6,9)(10,12)(13,15,19,32,20,31,18,30)(14,21,26,40,25,41,24,27)(16,34,17,29,36,28,35,42)(22,38,23,39,37,44,33,43);(5,7)(6,10)(8,11)(9,12)(13,16,20,36)(14,22,25,37)(15,28,31,34)(17,19,35,18)(21,38,41,44)(23,26,33,24)(27,39,40,43)(29,30,42,32);(2,4)(7,11)(10,12)(13,17,18,16,20,35,19,36)(14,23,24,22,25,33,26,37)(15,29,30,34,31,42,32,28)(21,39,27,38,41,43,40,44)

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=i,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=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, 366633);