#8926549

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	#56350	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4006 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833745);