#8505110

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	#42616	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4197 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=ghi,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, 8412306);