#7968665

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

4728 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,e^2=g,f^2=gh,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, 7875861);