#7689721

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4650 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)(17,18,23,25)(19,22,27,24)(20,21,26,32)(28,30,31,29);(1,2,5,6)(3,7,4,8)(9,11,14,13)(10,12,16,15)(17,19)(18,24)(20,28)(21,29)(22,25)(23,27)(26,31)(30,32);(1,3,5,4)(2,7,6,8)(9,11,14,13)(10,15,16,12)(18,25)(19,27)(20,26)(29,30);(9,12,14,15)(10,11,16,13)(17,20,23,26)(18,21,25,32)(19,28,27,31)(22,30,24,29);(9,13,14,11)(10,15,16,12)(17,21)(18,26)(19,29)(20,25)(22,28)(23,32)(24,31)(27,30);(1,4,5,3)(2,8,6,7)(17,22)(18,27)(19,25)(20,30)(21,31)(23,24)(26,29)(28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=h,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, 7596917);