#7826308

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

5028 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:

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

Polycyclic presentation:

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

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, 7733504);