#7674471

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4440 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

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, 7581667);