#7696242

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×D₄²	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5079 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7603438);