#8285573

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	#40365	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3989 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8192769);