#8728658

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	#36694	C₂
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)(3,8)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,34)(35,37)(36,40)(38,41)(39,43)(42,45)(44,48)(46,47);(2,7)(3,11)(9,12)(10,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,46)(45,48);(2,7)(3,11)(9,12)(10,16)(19,26)(22,23)(27,32)(30,31)(36,44)(40,48)(42,46)(45,47);(1,3,6,11)(2,8,7,13)(4,10,14,16)(5,9,15,12)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,36,38,44)(34,40,41,48)(35,42,43,46)(37,45,39,47);(1,4,6,14)(2,9,7,12)(3,10,11,16)(5,8,15,13)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,37,38,39)(34,35,41,43)(36,45,44,47)(40,42,48,46);(1,5,6,15)(2,10,7,16)(3,12,11,9)(4,13,14,8)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32)(33,35)(34,37)(36,46)(38,43)(39,41)(40,47)(42,44)(45,48)

Polycyclic presentation:

2^9|d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8635854);