#8908355

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	#38500	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815551);