#8425461

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	#40797	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4129 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332657);