#8475811

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	#45601	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4180 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)(3,10)(4,5)(6,7)(8,13)(9,14)(11,12)(15,16)(17,18)(19,25)(20,26)(21,24)(22,23)(27,31)(28,32)(29,30)(33,34)(35,42)(36,45)(37,39)(38,41)(40,46)(43,47)(44,48);(2,7)(4,14)(8,10)(11,16)(18,23)(19,26)(24,30)(27,32)(33,35,38,37)(34,39,41,42)(36,43,46,44)(40,47,45,48);(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,36)(34,40)(35,43)(37,44)(38,46)(39,47)(41,45)(42,48);(1,3,6,13)(2,8,7,10)(4,11,14,16)(5,15,9,12)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,36)(34,40)(35,44)(37,43)(38,46)(39,48)(41,45)(42,47);(1,4)(2,9)(3,11)(5,7)(6,14)(8,12)(10,15)(13,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,47)(45,46);(1,5,6,9)(2,4,7,14)(3,12,13,15)(8,16,10,11)(17,21,22,29)(18,24,23,30)(19,27,26,32)(20,28,25,31)(33,37,38,35)(34,39,41,42)(36,44,46,43)(40,47,45,48)

Polycyclic presentation:

2^9|b^2=hi,d^2=gh,f^2=g,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, 8383007);