#301999

order	512
nilpotency class	3
isoclinism	isoclinic class of #301993
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62269	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

992 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)(9,16)(10,14)(11,13)(12,15)(17,18)(19,24)(20,34)(21,27)(22,28)(23,31)(25,35)(26,29)(30,44)(32,38)(33,40)(36,41)(37,47)(39,45)(42,43)(46,48);(1,3)(2,6)(4,11)(5,10)(7,13)(8,14)(9,12)(15,16)(17,19,29,32,22,33,21,31)(18,23,27,40,28,38,26,24)(20,30,43,46,36,47,35,45)(25,37,41,48,42,44,34,39);(1,4)(2,7)(3,9,10,11)(5,12)(6,13,14,16)(8,15)(17,19,22,33)(18,24,28,40)(20,30,36,47)(21,31,29,32)(23,26,38,27)(25,39,42,48)(34,44,41,37)(35,45,43,46);(4,12)(7,15)(9,11)(13,16)(17,20,21,35,22,36,29,43)(18,25,26,34,28,42,27,41)(19,30,31,45,33,47,32,46)(23,37,24,39,38,44,40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209195);