#8811188

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	#53847	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=gh,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, 8718384);