#8818109

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=gi,e^2=gh,f^2=gi,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, 8725305);