#347894

order	512
nilpotency class	3
isoclinism	isoclinic class of #347878
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)(26,36)(27,37)(29,44)(30,31)(32,33)(35,39)(38,47)(41,42)(43,45);(17,18,24,37,22,27,21,23)(19,25,28,39,31,40,30,35)(20,26,33,41,34,42,32,36)(29,38,43,48,45,47,44,46);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,22,31)(18,25,27,40)(20,29,34,45)(21,30,24,28)(23,35,37,39)(26,38,42,47)(32,44,33,43)(36,46,41,48);(17,20)(18,26)(19,29)(21,32)(22,34)(23,36)(24,33)(25,38)(27,42)(28,43)(30,44)(31,45)(35,46)(37,41)(39,48)(40,47)

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,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, 255090);