#8810077

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	#47129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18,23,21)(19,24,29,31)(20,26,30,25)(22,32,28,27)(33,34,40,44)(35,37,38,41)(36,43,39,42)(45,46,48,47);(2,6)(7,9)(8,13)(14,15)(17,19,23,29)(18,24,21,31)(20,22,30,28)(25,27,26,32)(33,35)(34,41)(36,45)(37,44)(38,40)(39,48)(42,46)(43,47);(3,10)(4,12)(7,9)(8,13)(17,20,23,30)(18,25,21,26)(19,28,29,22)(24,32,31,27)(33,36,40,39)(34,42,44,43)(35,45,38,48)(37,47,41,46);(1,3,5,10)(2,7,6,9)(4,11,12,16)(8,14,13,15)(17,20)(18,26)(19,22)(21,25)(23,30)(24,32)(27,31)(28,29)(33,37,40,41)(34,35,44,38)(36,46,39,47)(42,48,43,45);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,21,23,18)(19,24,29,31)(20,25,30,26)(22,32,28,27)(33,38,40,35)(34,41,44,37)(36,45,39,48)(42,47,43,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,22)(18,27)(19,20)(21,32)(23,28)(24,25)(26,31)(29,30)(33,39,40,36)(34,43,44,42)(35,45,38,48)(37,47,41,46)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717273);