#9847669

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9754865);