#7658765

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

4756 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565961);