#399052

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

893 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=ef,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,g^c=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, 306248);