#8755002

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)(3,10)(4,12)(5,8)(6,7)(9,14)(11,15)(13,16)(17,18)(19,28)(20,21)(22,30)(23,24)(25,29)(26,32)(27,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,6)(8,10)(9,12)(15,16)(18,24)(19,29)(21,26)(27,30)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(1,3,7,5)(2,8,6,10)(4,13,14,11)(9,16,12,15)(17,19)(18,25)(20,30)(21,31)(22,26)(23,29)(24,28)(27,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,4,7,14)(2,9,6,12)(3,11,5,13)(8,15,10,16)(17,20)(18,26)(19,30)(21,24)(22,25)(23,32)(27,29)(28,31);(1,5,7,3)(2,10,6,8)(4,13,14,11)(9,16,12,15)(17,21,23,26)(18,20,24,32)(19,31,29,22)(25,30,28,27)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47);(1,6,7,2)(3,10,5,8)(4,12,14,9)(11,16,13,15)(17,22,23,31)(18,27,24,30)(19,26,29,21)(20,25,32,28)(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=gh,c^2=gi,d^2=gi,e^2=h,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, 8662198);