#8755013

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3771 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:

(1,2,6,7)(3,11,12,8)(4,14,15,9)(5,13,16,10)(17,18,23,24)(19,27,28,25)(20,31,32,21)(22,29,30,26)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,7)(5,16)(8,11)(9,14)(18,24)(19,28)(21,31)(26,29)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(2,7)(3,12)(9,14)(10,13)(17,19,23,28)(18,25,24,27)(20,29,32,26)(21,30,31,22)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,3,6,12)(2,8,7,11)(4,13,15,10)(5,9,16,14)(17,20)(18,21)(19,29)(22,27)(23,32)(24,31)(25,30)(26,28);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,21)(18,20)(19,30)(22,28)(23,31)(24,32)(25,29)(26,27)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47);(1,5,6,16)(2,10,7,13)(3,9,12,14)(4,11,15,8)(17,22,23,30)(18,26,24,29)(19,21,28,31)(20,27,32,25)(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=gh,c^2=i,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8662209);