#347922

order	512
nilpotency class	3
isoclinism	isoclinic class of #347908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
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

821 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^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, 255118);