#8156993

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	#40733	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4205 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18)(19,24)(20,26)(21,30)(22,29)(23,25)(27,31)(28,32)(36,44)(39,46)(42,48)(45,47);(2,7)(3,9)(10,13)(11,16)(17,19)(18,24)(20,23)(21,29)(22,30)(25,26)(27,32)(28,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(2,7)(3,9)(10,13)(11,16)(18,25)(19,20)(27,30)(28,29)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,20,23,19)(18,26,25,24)(21,29,31,28)(22,27,32,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,21)(18,27)(19,29)(20,28)(22,26)(23,31)(24,32)(25,30)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,22)(18,28)(19,30)(20,27)(21,26)(23,32)(24,31)(25,29)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)

Polycyclic presentation:

2^9|c^2=ghi,d^2=gh,e^2=g,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, 8064189);