#7897860

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4476 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)(3,7)(4,5)(6,8)(9,10,13,12)(11,15,16,14)(18,23)(19,26)(20,28)(29,30);(9,11,13,16)(10,14,12,15)(17,18,22,23)(19,24,26,21)(20,25,28,31)(27,29,32,30);(2,5)(6,7)(9,10,13,12)(11,14,16,15)(17,19,22,26)(18,21,23,24)(20,27,28,32)(25,30,31,29);(1,2)(3,6)(4,5)(7,8)(17,19)(18,24)(20,27)(21,23)(22,26)(25,29)(28,32)(30,31);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,15,16,14)(17,20)(18,25)(19,27)(21,30)(22,28)(23,31)(24,29)(26,32);(17,21,22,24)(18,19,23,26)(20,29,28,30)(25,32,31,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,e^2=gi,f^2=i,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, 7805056);