#7672426

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	#59059	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4668 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)(3,4)(5,8)(6,7)(10,13)(14,16)(18,24)(22,32)(25,28)(26,30);(9,10,12,13)(11,14,15,16)(17,18)(19,28)(20,25)(21,26)(22,27)(23,24)(29,32)(30,31);(2,7)(3,8)(17,19,23,20)(18,25,24,28)(21,27,31,29)(22,26,32,30);(1,3,6,8)(2,4,7,5)(9,11,12,15)(10,14,13,16)(17,20,23,19)(18,25,24,28)(21,27,31,29)(22,30,32,26);(1,4,6,5)(2,3,7,8)(17,21,23,31)(18,26,24,30)(19,29,20,27)(22,28,32,25);(1,5,6,4)(2,8,7,3)(17,22)(18,27)(19,26)(20,30)(21,28)(23,32)(24,29)(25,31)

Polycyclic presentation:

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

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, 7579622);