#7971704

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	#2846	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7878900);