#7746066

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4638 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13);(1,3)(2,4)(5,8)(6,7)(9,11,12,16)(10,13,14,15)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32);(2,6)(3,8)(18,22)(23,29)(24,27)(30,31);(1,3)(2,7)(4,6)(5,8)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31);(1,4,5,7)(2,3,6,8)(17,20)(18,24)(19,25)(21,28)(22,27)(23,30)(26,32)(29,31);(1,3)(2,7)(4,6)(5,8)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)

Polycyclic presentation:

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

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, 7653262);