#8505480

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	#53653	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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)(17,18,22,27)(19,26,30,23)(20,24,25,21)(28,31,32,29)(33,34)(35,40)(36,37)(38,41)(39,42)(43,47)(44,45)(46,48);(1,3)(2,8)(4,11)(5,6)(7,13)(9,15)(10,14)(12,16)(17,19)(18,23)(20,28)(21,31)(22,30)(24,29)(25,32)(26,27)(33,35)(34,40)(36,42)(37,39)(38,45)(41,44)(43,46)(47,48);(1,2,7,4)(3,11,13,8)(5,9,16,14)(6,10,12,15)(17,20,22,25)(18,24,27,21)(19,28,30,32)(23,29,26,31)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,48)(46,47);(1,4,7,2)(3,11,13,8)(5,14,16,9)(6,10,12,15)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,29)(31,32)(33,37,39,35)(34,36,42,40)(38,45,46,43)(41,44,48,47);(1,5)(2,9)(3,12)(4,14)(6,13)(7,16)(8,10)(11,15)(17,21,22,24)(18,25,27,20)(19,29,30,31)(23,28,26,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48);(1,6,7,12)(2,10,4,15)(3,5,13,16)(8,14,11,9)(17,19)(18,26)(20,28)(21,29)(22,30)(23,27)(24,31)(25,32)(33,36)(34,37)(35,42)(38,44)(39,40)(41,45)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,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, 8412676);