#9713758

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	#38280	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,e^2=h,f^2=g,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, 9620954);