#7711273

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	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7618469);