#8295525

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	#53455	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4091 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8202721);