#7655512

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	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5540 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)(3,7,8,5)(10,13)(11,15)(17,18,23,21)(19,26,24,20)(22,25,30,29)(27,32,31,28);(3,8)(5,7)(9,10,12,13)(11,14,15,16);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,32)(30,31);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,23,26)(18,24,21,19)(22,28,30,32)(25,31,29,27);(17,21,23,18)(19,26,24,20)(22,25,30,29)(27,28,31,32);(17,22,23,30)(18,25,21,29)(19,27,24,31)(20,28,26,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562708);