#432364

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

911 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,18,22,28)(19,25,32,41)(20,34,21,26)(23,29,24,39)(27,36,45,37)(30,46,47,40)(31,44,35,48)(33,42,38,43);(1,2,4,5)(3,6,7,8)(10,13)(11,15)(17,19)(18,25)(20,35)(21,31)(22,32)(23,38)(24,33)(26,44)(27,40)(28,41)(29,43)(30,36)(34,48)(37,47)(39,42)(45,46);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,22,21)(18,26,28,34)(19,30,32,47)(23,37,24,36)(25,40,41,46)(27,29,45,39)(31,38,35,33)(42,48,43,44);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,22,20)(18,27,28,45)(19,31,32,35)(23,37,24,36)(25,40,41,46)(26,29,34,39)(30,38,47,33)(42,44,43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=ef,d^2=f,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, 339560);