#428513

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

Polycyclic presentation:

2^9|a^2=f,b^2=ei,d^2=fi,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, 335709);