#7941741

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

4707 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7848937);