#7670696

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
prime radical	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4878 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,e^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577892);