#8496818

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	#45045	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4027 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8404014);