#7876719

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

5047 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)(4,7)(5,8)(10,13)(14,16)(17,18,23,21)(19,24,28,22)(20,25,31,29)(26,32,27,30);(1,3)(2,6)(4,8)(5,7)(9,10,12,13)(11,14,15,16)(17,19,23,28)(18,24,21,22)(20,27,31,26)(25,30,29,32);(1,2,5,8)(3,4,7,6)(17,20)(18,25)(19,26)(21,29)(22,30)(23,31)(24,32)(27,28);(1,3)(2,6)(4,8)(5,7)(17,21,23,18)(19,22,28,24)(20,29,31,25)(26,30,27,32);(1,3)(2,4)(5,7)(6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,27)(21,29)(22,32)(23,31)(24,30)(26,28);(1,4)(2,7)(3,8)(5,6)(17,22)(18,19)(20,30)(21,28)(23,24)(25,26)(27,29)(31,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=hi,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7783915);