#347919

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	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

815 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)(21,33)(24,34)(25,35)(26,36)(27,40)(28,43)(30,44)(31,39)(37,47)(38,46)(41,45);(17,18,21,23,22,26,33,36)(19,24,28,34,29,38,43,46)(20,30,31,44,32,25,39,35)(27,41,42,47,40,37,48,45);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,24)(20,27)(21,28)(22,29)(23,34)(25,37)(26,38)(30,41)(31,42)(32,40)(33,43)(35,45)(36,46)(39,48)(44,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20)(18,25)(19,27)(21,31)(22,32)(23,35)(24,37)(26,30)(28,42)(29,40)(33,39)(34,45)(36,44)(38,41)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=e,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, 255115);