#7648733

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	#37673	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4979 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)(19,26)(20,30)(21,29)(22,24)(23,25)(27,31)(28,32);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,21)(20,27)(22,28)(23,29)(24,31)(25,26)(30,32);(2,5)(6,8)(17,20,23,22)(18,24,25,30)(19,27,29,28)(21,31,26,32);(3,7)(6,8)(17,21)(18,19)(20,31)(22,32)(23,26)(24,27)(25,29)(28,30);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,22,23,20)(18,24,25,30)(19,28,29,27)(21,31,26,32);(1,2)(3,6)(4,5)(7,8)(17,18)(19,21)(20,30)(22,24)(23,25)(26,29)(27,32)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7555929);