#9058915

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	#35827	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4070 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8966111);