#8322127

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3870 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)(3,5)(4,12)(6,7)(8,10)(9,15)(11,16)(13,14)(33,34)(35,40)(36,45)(37,38)(39,42)(41,48)(43,47)(44,46);(2,7)(3,10)(9,12)(11,13)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(33,35,39,36)(34,40,42,45)(37,46,41,43)(38,44,48,47);(2,7)(5,8)(9,12)(14,16)(18,23)(21,31)(24,29)(25,32)(33,36,39,35)(34,40,42,45)(37,46,41,43)(38,47,48,44);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,15)(12,16)(17,19)(18,24)(20,26)(21,25)(22,28)(23,29)(27,30)(31,32)(33,37,39,41)(34,38,42,48)(35,43,36,46)(40,47,45,44);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,32)(24,31)(26,28)(33,38)(34,41)(35,44)(36,47)(37,42)(39,48)(40,43)(45,46);(1,5,6,8)(2,3,7,10)(4,14,15,16)(9,11,12,13)(17,21,22,31)(18,26,23,27)(19,25,28,32)(20,29,30,24)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=gi,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, 8229323);