#8755122

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3769 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,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, 8662318);