#7745344

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

5357 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)(3,4)(5,6)(7,8)(9,10,14,12)(11,16,13,15)(17,18)(19,27)(20,24)(21,32)(22,26)(23,25)(28,31)(29,30);(9,11,14,13)(10,15,12,16)(17,19,23,20)(18,24,25,27)(21,30,26,28)(22,29,32,31);(2,6)(3,7)(10,12)(15,16)(18,25)(19,20)(22,32)(28,30);(1,3,5,7)(2,4,6,8)(9,12,14,10)(11,16,13,15)(17,20,23,19)(18,24,25,27)(21,30,26,28)(22,31,32,29);(1,4)(2,7)(3,6)(5,8)(9,13,14,11)(10,16,12,15)(17,21,23,26)(18,22,25,32)(19,28,20,30)(24,31,27,29);(17,22)(18,26)(19,29)(20,31)(21,25)(23,32)(24,28)(27,30)

Polycyclic presentation:

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

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, 7652540);