#7654008

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	#42438	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4922 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,e^2=g,f^2=g,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, 7561204);