#8107292

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	#52697	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=hi,e^2=gh,f^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

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, 8014488);