#8183854

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	#34962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8091050);