#8810209

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	#44318	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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)(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31);(2,6)(7,9)(8,13)(14,15)(18,23)(20,25)(21,30)(26,32)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(3,10)(4,12)(7,9)(8,13)(17,19,22,27)(18,20,23,25)(21,26,30,32)(24,28,29,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,3,5,10)(2,7,6,9)(4,11,12,16)(8,14,13,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=gh,e^2=gh,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, 8717405);