#7689751

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

4758 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)(17,18,23,29)(19,22,28,24)(20,26,31,25)(21,27,30,32);(1,2,5,6)(3,7,4,8)(9,11,14,13)(10,12,16,15)(17,19)(18,24)(20,21)(22,29)(23,28)(25,27)(26,32)(30,31);(9,11,14,13)(10,15,16,12)(17,20,23,31)(18,25,29,26)(19,30,28,21)(22,27,24,32);(1,3,5,4)(2,7,6,8)(9,12,14,15)(10,11,16,13)(17,20)(18,26)(19,21)(22,27)(23,31)(24,32)(25,29)(28,30);(9,13,14,11)(10,15,16,12)(17,21)(18,27)(19,20)(22,26)(23,30)(24,25)(28,31)(29,32);(1,4,5,3)(2,8,6,7)(17,22)(18,28)(19,29)(20,27)(21,25)(23,24)(26,30)(31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,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, 7596947);