#7718249

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4992 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:

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

Polycyclic presentation:

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

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, 7625445);