#7710924

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	#45483	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4846 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618120);