#8471201

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5505 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:

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

Polycyclic presentation:

2^9|d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8378397);