#7642349

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	#38356	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4921 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=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, 7549545);