#8295542

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8202738);