#7719920

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	#34624	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4710 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:

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

Polycyclic presentation:

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

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, 7627116);