#8112552

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	#56722	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8019748);