#7641908

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	#37546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4852 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:

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

Polycyclic presentation:

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

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, 7549104);