#7656723

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	#42438	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4797 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7563919);