#323202

order	512
nilpotency class	3
isoclinism	isoclinic class of #323197
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3151	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

675 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230398);