#7710931

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	#34984	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4998 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)(3,4)(5,6)(7,8)(10,13)(14,16)(17,18)(19,25)(20,23)(21,29)(22,24)(26,30)(27,31)(28,32);(9,10,12,13)(11,14,15,16)(17,19,22,20)(18,23,24,25)(21,27,30,28)(26,31,29,32);(2,6)(3,7)(18,24)(19,20)(26,29)(27,28);(1,3,5,7)(2,4,6,8)(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,32)(29,30);(1,4)(2,7)(3,6)(5,8)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31);(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)

Polycyclic presentation:

2^9|b^2=g,d^2=gh,e^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, 7618127);