#8528875

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	#54931	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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)(3,8)(4,11)(5,9)(6,7)(10,16)(12,13)(14,15)(18,22)(19,25)(24,29)(26,31)(33,34,38,42)(35,37,41,39)(36,44,46,40)(43,47,48,45);(1,2)(3,4)(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,47,44,45);(2,7)(4,8)(9,15)(12,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,38,46)(34,40,42,44)(35,43,41,48)(37,45,39,47);(1,3)(2,8)(4,7)(5,13)(6,11)(9,12)(10,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,47)(44,45)(46,48);(1,4,6,8)(2,3,7,11)(5,12,14,16)(9,10,15,13)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,48,44,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,e^2=gh,f^2=g,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, 8436071);