#7738218

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	#35237	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4984 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7645414);