#8096991

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	#52985	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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)(3,6)(4,8)(5,9)(7,12)(10,14)(11,15)(13,16)(17,18,24,23)(19,21,28,25)(20,29,22,26)(27,32,30,31)(34,40)(36,41)(42,48)(43,46);(1,3,5,11)(2,6,9,15)(4,10,12,16)(7,13,8,14)(17,19)(18,25)(20,30)(21,23)(22,27)(24,28)(26,31)(29,32)(33,34,39,40)(35,41,45,36)(37,42,44,48)(38,43,47,46);(1,4)(2,7)(3,10)(5,12)(6,13)(8,9)(11,16)(14,15)(17,20,24,22)(18,26,23,29)(19,27,28,30)(21,31,25,32)(33,35,39,45)(34,41,40,36)(37,38,44,47)(42,43,48,46);(1,3)(2,6)(4,10)(5,11)(7,13)(8,14)(9,15)(12,16)(17,21)(18,19)(20,31)(22,32)(23,28)(24,25)(26,27)(29,30)(33,36)(34,35)(37,46)(38,42)(39,41)(40,45)(43,44)(47,48);(1,4)(2,8)(3,10)(5,12)(6,14)(7,9)(11,16)(13,15)(17,22,24,20)(18,26,23,29)(19,27,28,30)(21,32,25,31)(33,37)(34,42)(35,38)(36,46)(39,44)(40,48)(41,43)(45,47);(1,2)(3,6)(4,7)(5,9)(8,12)(10,13)(11,15)(14,16)(17,23,24,18)(19,21,28,25)(20,29,22,26)(27,31,30,32)(33,38)(34,43)(35,44)(36,42)(37,45)(39,47)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004187);