#428510

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1197 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)(5,6,8,9)(7,10,11,12)(13,14)(15,25)(16,21)(17,23)(18,22)(19,24)(20,27)(26,41)(28,35)(29,36)(30,44)(31,38)(32,40)(33,39)(34,42)(37,43);(1,2,3,4)(13,15,17,27,19,29,18,28)(14,20,22,25,24,35,23,36)(16,30,31,43,32,42,33,26)(21,37,39,44,40,41,38,34);(6,9)(10,12)(13,16)(14,21)(15,26)(17,33)(18,31)(19,32)(20,34)(22,38)(23,39)(24,40)(25,41)(27,42)(28,30)(29,43)(35,44)(36,37);(2,4)(5,7,8,11)(6,10,9,12)(14,22)(15,27)(17,18)(20,35)(21,38)(23,24)(26,42)(28,29)(30,43)(31,33)(34,44)(39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=f,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^b=dei,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335706);