#301986

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,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, 209182);