#9945515

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	#58320	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3970 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9852711);