#8108777

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

3986 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015973);