#8425519

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	#39996	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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)(20,24)(25,31)(28,32)(33,34)(35,38)(36,47)(37,39)(40,45)(41,44)(42,46)(43,48);(1,3,8,6)(2,9,7,12)(4,14,5,13)(10,16,11,15)(18,23)(19,26)(25,31)(27,29)(33,35,40,44)(34,41,45,38)(36,39,46,43)(37,47,48,42);(1,4,8,5)(2,10,7,11)(3,13,6,14)(9,15,12,16)(17,18,22,23)(19,24,26,20)(21,25,30,31)(27,32,29,28)(33,36)(34,42)(35,39)(37,41)(38,48)(40,46)(43,44)(45,47);(1,5,8,4)(2,11,7,10)(3,14,6,13)(9,16,12,15)(17,19)(18,24)(20,23)(21,29)(22,26)(25,28)(27,30)(31,32)(33,37,40,48)(34,43,45,39)(35,42,44,47)(36,41,46,38);(1,6,8,3)(2,12,7,9)(4,14,5,13)(10,16,11,15)(17,20,22,24)(18,19,23,26)(21,28,30,32)(25,27,31,29)(33,38,40,41)(34,44,45,35)(36,48,46,37)(39,42,43,47);(1,7,8,2)(3,12,6,9)(4,11,5,10)(13,16,14,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,39,40,43)(34,37,45,48)(35,46,44,36)(38,42,41,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=g,e^2=h,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, 8332715);