#8908128

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

4047 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=gh,e^2=gh,f^2=h,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, 8815324);