#7685067

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	#45057	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4930 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592263);