#7805063

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	#58995	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4919 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)(3,7)(4,5)(6,8)(10,13)(14,16)(17,18,23,21)(19,27,28,24)(20,25,31,30)(22,26,29,32);(1,2,4,5)(3,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,27)(22,31)(23,28)(25,26)(30,32);(2,5)(6,7)(19,28)(22,29)(24,27)(26,32);(1,2)(3,6)(4,5)(7,8)(17,20)(18,25)(19,22)(21,30)(23,31)(24,32)(26,27)(28,29);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,14,13,16)(17,21,23,18)(19,24,28,27)(20,30,31,25)(22,32,29,26);(1,2)(3,6)(4,5)(7,8)(17,22)(18,26)(19,20)(21,32)(23,29)(24,30)(25,27)(28,31)

Polycyclic presentation:

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

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, 7712259);