#331001

order	512
nilpotency class	3
isoclinism	isoclinic class of #330975
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

491 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238197);