#348514

order	512
nilpotency class	3
isoclinism	isoclinic class of #348478
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

866 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)(3,10)(4,8)(5,7)(6,11)(9,13)(12,15)(14,16)(17,18,23,26)(19,22,29,40)(20,30,34,27)(21,35,39,25)(24,32,42,33)(28,38,36,37)(31,44,46,43)(41,45,48,47);(1,3)(2,6)(4,11)(5,12)(7,14)(8,10)(9,15)(13,16)(17,19,22,26,23,29,40,18)(20,27,32,42,34,30,33,24)(21,36,37,25,39,28,38,35)(31,44,45,41,46,43,47,48);(2,7)(5,13)(6,14)(8,9)(10,15)(12,16)(17,20)(18,24)(19,27)(21,31)(22,32)(23,34)(25,41)(26,42)(28,43)(29,30)(33,40)(35,48)(36,44)(37,45)(38,47)(39,46);(3,11)(6,10)(12,16)(14,15)(17,21)(18,25)(19,28)(20,31)(22,37)(23,39)(24,41)(26,35)(27,43)(29,36)(30,44)(32,45)(33,47)(34,46)(38,40)(42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=di,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255710);