#8755009

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)(3,10,13,8)(4,5,14,9)(11,15,16,12)(17,18,22,23)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,7)(5,9)(8,10)(12,15)(18,23)(20,28)(24,26)(27,29)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(1,3,6,13)(2,8,7,10)(4,11,14,16)(5,12,9,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,3)(2,8)(4,11)(5,12)(6,13)(7,10)(9,15)(14,16)(17,19,22,21)(18,24,23,26)(20,27,28,29)(25,31,30,32);(1,4)(2,9)(3,11)(5,7)(6,14)(8,15)(10,12)(13,16)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47);(1,5,6,9)(2,4,7,14)(3,12,13,15)(8,11,10,16)(17,21,22,19)(18,26,23,24)(20,29,28,27)(25,32,30,31)(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=gi,d^2=i,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, 8662205);