#8003806

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

5000 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)(3,6)(4,5)(7,8)(10,14)(15,16)(17,18,22,23)(19,24,27,31)(20,25,28,32)(21,29,30,26);(1,3,5,8)(2,6,4,7)(9,10,13,14)(11,15,12,16)(18,23)(20,28)(24,31)(26,29);(1,3)(2,7)(4,6)(5,8)(9,11,13,12)(10,15,14,16)(18,23)(19,27)(25,32)(26,29);(1,3,5,8)(2,6,4,7)(9,11,13,12)(10,15,14,16)(17,19)(18,24)(20,26)(21,25)(22,27)(23,31)(28,29)(30,32);(1,3)(2,6)(4,7)(5,8)(17,20)(18,25)(19,26)(21,24)(22,28)(23,32)(27,29)(30,31);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,16,14,15)(17,21)(18,26)(19,25)(20,24)(22,30)(23,29)(27,32)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7911002);