#7696464

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	#39942	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4603 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7603660);