#431089

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1113 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)(5,6,9,11)(7,12,8,10)(13,14,19,25)(15,26,17,34)(16,29,33,22)(18,36,20,24)(21,35,23,28)(27,43,31,39)(30,38,44,40)(32,41,37,42);(1,2,3,4)(13,15,20,35,19,17,18,28)(14,21,36,34,25,23,24,26)(16,30,37,31,33,44,32,27)(22,39,42,38,29,43,41,40);(5,7,9,8)(6,10,11,12)(13,16)(14,22)(15,27)(17,31)(18,32)(19,33)(20,37)(21,38)(23,40)(24,41)(25,29)(26,39)(28,44)(30,35)(34,43)(36,42);(2,4)(5,8,9,7)(6,10,11,12)(13,17,19,15)(14,23,25,21)(16,31,33,27)(18,28,20,35)(22,40,29,38)(24,26,36,34)(30,32,44,37)(39,42,43,41)

Polycyclic presentation:

2^9|a^2=f,b^2=ei,c^2=fi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei,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, 338285);