#7754826

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

4924 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:

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

Polycyclic presentation:

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

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, 7662022);