#8534623

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	#40725	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4237 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8441819);