#8154770

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

4203 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=g,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, 8061966);