#347924

order	512
nilpotency class	3
isoclinism	isoclinic class of #347908
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

792 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)(3,9,11,16)(4,12,14,7)(6,13,15,10)(18,23)(19,28)(21,24)(26,37)(27,38)(29,44)(30,31)(32,42)(33,34)(36,40)(39,45)(43,46);(17,18,24,38,22,27,21,23)(19,25,28,40,31,41,30,36)(20,32,34,37,35,26,33,42)(29,45,43,47,46,39,44,48);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,22,31)(18,25,27,41)(20,29,35,46)(21,30,24,28)(23,36,38,40)(26,39,32,45)(33,44,34,43)(37,47,42,48);(17,20)(18,26)(19,29)(21,33)(22,35)(23,37)(24,34)(25,39)(27,32)(28,43)(30,44)(31,46)(36,47)(38,42)(40,48)(41,45)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255120);