#8504812

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,d^2=i,e^2=h,f^2=hi,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, 8412008);