#8504928

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3886 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=i,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, 8412124);