#7633300

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	#35238	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5100 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540496);