#7670956

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	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4679 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)(18,23)(21,31)(24,30)(25,32);(1,2,5,6)(3,7,4,8)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32);(9,11,14,13)(10,15,16,12)(18,23)(20,29)(24,30)(26,27);(1,3,5,4)(2,7,6,8)(9,12,14,15)(10,11,16,13)(17,19,22,28)(18,24,23,30)(20,27,29,26)(21,25,31,32);(9,13,14,11)(10,15,16,12)(17,20)(18,25)(19,27)(21,30)(22,29)(23,32)(24,31)(26,28);(1,4,5,3)(2,8,6,7)(17,21,22,31)(18,26,23,27)(19,25,28,32)(20,30,29,24)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=gh,e^2=h,f^2=g,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, 7578152);