#7656941

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	#42950	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4733 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=g,e^2=ghi,f^2=hi,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, 7564137);