#7876333

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	#37833	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5267 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=hi,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7783529);