#234889

order	512
nilpotency class	3
isoclinism	isoclinic class of #234878
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1220 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=di,e^b=eh,d^c=dh,e^c=ei,f^c=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, 142085);