#8375652

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	#59677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3992 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,14)(15,16)(17,18,23,27)(19,24,28,31)(20,21,25,32)(22,26,29,30)(34,37)(35,36)(41,45)(43,44);(2,7)(4,8)(9,14)(12,15)(17,19,23,28)(18,24,27,31)(20,29,25,22)(21,30,32,26)(33,34)(35,40)(36,42)(37,39)(38,41)(43,47)(44,48)(45,46);(5,13)(9,14)(10,16)(12,15)(19,28)(22,29)(24,31)(26,30)(33,35,39,36)(34,40,37,42)(38,44,46,43)(41,48,45,47);(1,3,6,11)(2,8,7,4)(5,10,13,16)(9,15,14,12)(17,20)(18,21)(19,29)(22,28)(23,25)(24,30)(26,31)(27,32)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,21)(18,25)(19,26)(20,27)(22,31)(23,32)(24,29)(28,30)(33,37,39,34)(35,42,36,40)(38,45,46,41)(43,48,44,47);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,22,23,29)(18,26,27,30)(19,20,28,25)(21,31,32,24)(33,38,39,46)(34,41,37,45)(35,43,36,44)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8282848);