#8755012

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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,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, 8662208);