#9712876

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9620072);