#7630599

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	#35567	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4966 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)(9,10,13,12)(11,15,16,14)(17,18,22,25)(19,23,27,20)(21,29,30,24)(26,28,32,31);(1,2)(3,6)(4,7)(5,8)(9,11,13,16)(10,14,12,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32);(1,2)(3,6)(4,7)(5,8)(17,19,22,27)(18,20,25,23)(21,26,30,32)(24,28,29,31);(1,3,5,4)(2,6,8,7)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32);(1,3,5,4)(2,6,8,7)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32);(1,4,5,3)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,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, 7537795);