#8417026

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8324222);