#8351625

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4041 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8258821);