#268092

order	512
nilpotency class	3
isoclinism	isoclinic class of #268043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#553	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

830 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:

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

Polycyclic presentation:

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

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, 175288);