#8471848

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4203 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8379044);