#424171

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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

1276 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331367);