#8094764

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

5076 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8001960);