#431571

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

917 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)(7,10)(8,11)(9,12)(13,14,18,23)(15,26,17,20)(16,21,32,39)(19,24,35,25)(22,29,37,28)(27,42,31,36)(30,44,43,38)(33,40,34,41);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,30)(17,18)(19,29)(21,38)(22,24)(23,26)(25,37)(27,33)(28,35)(31,34)(32,43)(36,40)(39,44)(41,42);(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,27)(17,31)(18,32)(19,34)(20,36)(22,38)(23,39)(24,41)(25,40)(26,42)(28,43)(29,30)(33,35)(37,44);(2,4)(5,8,9,7)(6,11,12,10)(13,17,18,15)(14,22,23,37)(16,31,32,27)(19,29,35,28)(20,25,26,24)(21,38,39,44)(30,33,43,34)(36,40,42,41)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=f,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=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, 338767);