#7942174

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4506 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,14,12,15)(17,18,22,23)(19,20,25,28)(21,24,29,30)(26,27,31,32);(1,2,6,4)(3,8,5,7)(9,10)(11,14)(12,15)(13,16);(1,3,6,5)(2,7,4,8)(18,23)(19,25)(24,30)(26,31);(1,2,6,4)(3,8,5,7)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32);(1,4,6,2)(3,8,5,7)(9,11,13,12)(10,14,16,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31);(1,5,6,3)(2,7,4,8)(9,12,13,11)(10,15,16,14)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,27,28,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=i,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7849370);