#7671206

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4768 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7578402);