#254532

order	512
nilpotency class	3
isoclinism	isoclinic class of #254515
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

826 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,22,26)(19,23,29,38)(20,30,31,24)(21,33,35,25)(27,40,34,36)(28,37,42,43)(32,39,45,44)(41,48,46,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(19,27,29,34)(23,36,38,40)(28,41,42,46)(37,47,43,48);(6,12)(13,16)(17,19)(18,23)(20,28)(21,34)(22,29)(24,37)(25,40)(26,38)(27,35)(30,43)(31,42)(32,46)(33,36)(39,48)(41,45)(44,47);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,20,22,31)(18,24,26,30)(19,28,29,42)(21,32,35,45)(23,37,38,43)(25,39,33,44)(27,41,34,46)(36,47,40,48)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=hi,g^2=h,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,g^c=gh,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, 161728);