#7819916

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4548 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7727112);