#8324111

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3871 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)(3,12)(4,14)(5,6)(7,9)(8,13)(10,16)(11,15)(17,18,23,26)(19,22,25,31)(20,21,24,30)(27,28,29,32)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,3,7,13)(2,8,9,12)(4,15,16,5)(6,10,11,14)(17,19)(18,22)(20,27)(21,28)(23,25)(24,29)(26,31)(30,32)(34,35)(39,41)(40,42)(46,47);(2,9)(6,11)(8,12)(10,14)(19,25)(22,31)(27,29)(28,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,7,16)(2,10,9,14)(3,5,13,15)(6,12,11,8)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,30)(31,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,5)(2,11)(3,4)(6,9)(7,15)(8,14)(10,12)(13,16)(17,21)(18,24)(19,28)(20,26)(22,29)(23,30)(25,32)(27,31)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47);(1,6,7,11)(2,5,9,15)(3,10,13,14)(4,12,16,8)(17,22,23,31)(18,25,26,19)(20,28,24,32)(21,29,30,27)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=h,e^2=gh,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, 8231307);