#285128

order	512
nilpotency class	3
isoclinism	isoclinic class of #285115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#525
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

597 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192324);