#8479498

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	#36791	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=gh,e^2=g,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, 8386694);