#8438056

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	#37496	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8345252);