#330974

order	512
nilpotency class	3
isoclinism	isoclinic class of #330923
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

552 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=hi,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238170);