#7735704

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

4822 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)(9,10,13,11)(12,15,16,14)(17,18,22,21)(19,24,26,23)(20,25,28,29)(27,31,32,30);(1,2)(3,6)(4,7)(5,8)(17,19)(18,23)(20,27)(21,24)(22,26)(25,30)(28,32)(29,31);(10,11)(14,15)(19,26)(20,28)(23,24)(25,29);(1,3,5,4)(2,6,8,7)(9,11,13,10)(12,15,16,14)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,32)(29,30);(1,3,5,4)(2,6,8,7)(9,12,13,16)(10,14,11,15)(17,20,22,28)(18,25,21,29)(19,27,26,32)(23,30,24,31);(1,4,5,3)(2,7,8,6)(17,21,22,18)(19,23,26,24)(20,29,28,25)(27,30,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7642900);