#8751695

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

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

5175 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658891);