#8505516

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	#48429	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=i,e^2=g,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, 8412712);