#9039359

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	#55595	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946555);