#330987

order	512
nilpotency class	3
isoclinism	isoclinic class of #330975
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2619	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

521 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=h,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, 238183);