#299252

order	512
nilpotency class	3
isoclinism	isoclinic class of #299233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

831 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206448);