#7630596

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	#35568	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4822 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537792);