#8097744

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	#52307	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4617 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004940);