#301988

order	512
nilpotency class	3
isoclinism	isoclinic class of #301973
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	#3536	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

974 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)(3,9)(4,7)(5,8)(6,11)(10,14)(12,16)(13,15)(17,18,23,28)(19,25,31,40)(20,32,36,47)(21,37,22,27)(24,38,41,30)(26,35,44,34)(29,45,33,39)(42,48,43,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,14,15,16)(17,19,22,38,23,31,21,30)(18,24,37,40,28,41,27,25)(20,29,35,48,36,33,34,46)(26,39,47,43,44,45,32,42);(1,4)(2,7)(3,10,11,12)(5,13)(6,14,9,16)(8,15)(17,19)(18,25)(20,33)(21,30)(22,38)(23,31)(24,37)(26,43)(27,41)(28,40)(29,36)(32,39)(34,48)(35,46)(42,44)(45,47);(2,8)(6,9)(7,15)(14,16)(17,20,21,34,23,36,22,35)(18,26,27,32,28,44,37,47)(19,29,30,46,31,33,38,48)(24,39,25,42,41,45,40,43)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,d^2=ei,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=dh,g^c=ghi

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, 209184);