#288714

order	512
nilpotency class	3
isoclinism	isoclinic class of #288683
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

570 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)(3,9)(4,6)(5,8)(7,11)(10,15)(12,14)(13,16)(17,18,21,24,23,28,22,27)(19,29,30,25,33,40,32,41)(20,34,36,26,35,38,37,42)(31,39,44,47,46,48,45,43);(2,6)(3,10)(5,12)(7,15)(9,16)(11,13)(18,24,28,27)(19,30,33,32)(20,35)(26,34,42,38)(31,45,46,44)(36,37)(39,48)(43,47);(1,3,4,11)(2,7,6,9)(5,13,12,10)(8,15,14,16)(17,19)(18,25)(20,31)(21,32)(22,30)(23,33)(24,29)(26,39)(27,40)(28,41)(34,47)(35,46)(36,45)(37,44)(38,43)(42,48);(2,6)(3,11)(8,14)(10,13)(17,20,22,37,23,35,21,36)(18,26,24,38,28,42,27,34)(19,31,30,44,33,46,32,45)(25,39,29,43,41,48,40,47)

Polycyclic presentation:

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

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, 195910);