#8810106

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	#45536	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=h,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, 8717302);