#7901423

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×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4515 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=hi,e^2=i,f^2=i,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, 7808619);