#8095847

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	#45914	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4109 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)(3,8)(4,7)(5,9)(6,11)(10,15)(12,14)(13,16)(18,23)(24,25)(26,30)(31,32)(33,34)(35,36)(37,45)(38,40)(39,42)(41,48)(43,47)(44,46);(4,12)(7,14)(10,16)(13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35)(34,38)(36,42)(37,46)(39,40)(41,44)(43,48)(45,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,20)(18,24,23,25)(21,27,29,28)(26,31,30,32)(33,36,39,38)(34,40,42,35)(37,44,48,47)(41,46,45,43);(1,4)(2,7)(3,10)(5,12)(6,13)(8,15)(9,14)(11,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,37)(34,41)(35,43)(36,44)(38,47)(39,48)(40,46)(42,45);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,16)(13,14)(17,20,22,19)(18,25,23,24)(21,27,29,28)(26,31,30,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,47)(46,48);(1,2)(3,6)(4,7)(5,9)(8,11)(10,13)(12,14)(15,16)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,38,39,36)(34,40,42,35)(37,47,48,44)(41,46,45,43)

Polycyclic presentation:

2^9|c^2=g,e^2=gi,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, 8003043);