#8712839

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8620035);