#331002

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

Polycyclic presentation:

2^9|a^2=hi,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, 238198);