#8657195

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	#47929	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4888 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=gi,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8564391);