#7684659

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	#43112	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4767 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)(11,12)(13,15)(14,16)(17,18)(19,28)(20,22)(21,26)(23,27)(24,31)(25,32)(29,30);(1,2,5,6)(3,7,4,8)(17,19)(18,24)(20,21)(22,30)(23,31)(25,26)(27,28)(29,32);(9,11,14,13)(10,15,16,12)(17,20,23,32)(18,25,27,22)(19,29,31,21)(24,30,28,26);(1,3,5,4)(2,7,6,8)(9,12,14,15)(10,11,16,13)(17,20)(18,22)(19,21)(23,32)(24,30)(25,27)(26,28)(29,31);(9,13,14,11)(10,15,16,12)(17,21)(18,26)(19,20)(22,28)(23,29)(24,25)(27,30)(31,32);(1,4,5,3)(2,8,6,7)(17,22)(18,20)(19,30)(21,24)(23,25)(26,31)(27,32)(28,29)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=gh,e^2=hi,f^2=gi,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, 7591855);