#7966949

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×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4617 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7874145);