#7677943

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	#45265	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4588 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585139);