#8269729

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

4072 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=ghi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8176925);