#8696708

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	#44326	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4178 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)(17,18,22,21)(19,24,27,23)(20,28,29,25)(26,30,32,31)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(1,3,8,6)(2,9,7,12)(4,14,5,13)(10,16,11,15)(17,18,22,21)(19,24,27,23)(20,25,29,28)(26,31,32,30)(35,44)(36,45)(40,46)(41,48);(1,4,8,5)(2,10,7,11)(3,13,6,14)(9,15,12,16)(17,19)(18,23)(20,26)(21,24)(22,27)(25,30)(28,31)(29,32)(33,35,39,44)(34,40,43,46)(36,38,45,47)(37,41,42,48);(1,5,8,4)(2,11,7,10)(3,14,6,13)(9,16,12,15)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,30)(29,32)(33,36)(34,41)(35,38)(37,46)(39,45)(40,42)(43,48)(44,47);(1,6,8,3)(2,12,7,9)(4,14,5,13)(10,16,11,15)(17,20)(18,25)(19,26)(21,28)(22,29)(23,30)(24,31)(27,32)(33,37)(34,38)(35,41)(36,46)(39,42)(40,45)(43,47)(44,48);(1,7,8,2)(3,12,6,9)(4,11,5,10)(13,16,14,15)(17,21,22,18)(19,24,27,23)(20,25,29,28)(26,30,32,31)(33,38)(34,42)(35,45)(36,44)(37,43)(39,47)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603904);