#8324388

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3840 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,13)(14,16)(17,18,23,26)(19,22,25,31)(20,21,24,30)(27,28,29,32)(33,34,38,42)(35,39,37,41)(36,40,45,48)(43,46,44,47);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,13,10,16)(17,19)(18,22)(20,27)(21,28)(23,25)(24,29)(26,31)(30,32)(33,35,38,37)(34,39,42,41)(36,44,45,43)(40,47,48,46);(2,9)(6,10)(8,11)(13,16)(19,25)(22,31)(27,29)(28,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4)(2,10)(3,12)(5,14)(6,9)(7,15)(8,13)(11,16)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,30)(31,32)(33,36,38,45)(34,40,42,48)(35,43,37,44)(39,46,41,47);(1,5,7,3)(2,11,9,8)(4,12,15,14)(6,16,10,13)(17,21)(18,24)(19,28)(20,26)(22,29)(23,30)(25,32)(27,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,14,11,12)(17,22,23,31)(18,25,26,19)(20,28,24,32)(21,29,30,27)(33,37,38,35)(34,41,42,39)(36,44,45,43)(40,47,48,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=gh,e^2=g,f^2=hi,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, 8231584);