#7847210

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

5612 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,e^2=i,f^2=i,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, 7754406);