#7763420

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	#37614	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5000 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)(3,4)(5,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18,22,21)(19,23,25,27)(20,28,30,24)(26,32,29,31);(9,11,13,16)(10,14,12,15)(17,19,22,25)(18,23,21,27)(20,29,30,26)(24,32,28,31);(2,6)(3,7)(9,12,13,10)(11,15,16,14)(18,21)(19,25)(24,28)(26,29);(1,3,5,7)(2,4,6,8)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30);(1,4)(2,7)(3,6)(5,8)(17,20,22,30)(18,24,21,28)(19,26,25,29)(23,31,27,32);(17,21,22,18)(19,27,25,23)(20,28,30,24)(26,32,29,31)

Polycyclic presentation:

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

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, 7670616);