#7711936

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	#36410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4804 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,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, 7619132);