#7942173

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4492 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7849369);