#8701697

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	C₂×#547	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4622 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8608893);