#7739457

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

5178 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)(11,15)(12,13)(14,16)(17,18)(19,22)(20,24)(21,25)(23,26)(27,29)(28,31)(30,32);(1,2)(3,6)(4,7)(5,8)(17,19)(18,22)(20,27)(21,26)(23,25)(24,29)(28,32)(30,31);(1,3,5,4)(2,6,8,7)(10,13)(14,15)(19,26)(20,28)(22,23)(24,31);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32);(1,4,5,3)(2,7,8,6)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

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, 7646653);