#8828871

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

5161 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(2,3)(4,8)(9,10,14,12)(11,15,13,16)(17,18,22,24)(19,20,27,23)(21,29,31,25)(26,32,30,28);(2,3)(4,8)(9,11,14,13)(10,15,12,16)(17,19,22,27)(18,23,24,20)(21,30,31,26)(25,28,29,32);(1,2,5,3)(4,6,8,7)(9,11,14,13)(10,16,12,15)(18,24)(20,23)(25,29)(28,32);(2,3)(6,7)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,30)(31,32);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,13,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,32)(28,29)(30,31);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,15,12,16)(17,21,22,31)(18,25,24,29)(19,26,27,30)(20,28,23,32)

Polycyclic presentation:

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

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, 8736067);