#8036930

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

4832 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7944126);