#8158674

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	#53525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=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, 8065870);