#253728

order	512
nilpotency class	3
isoclinism	isoclinic class of #253699
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	#3505	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

783 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160924);