#8186933

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
prime radical	#39383	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4818 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)(11,12)(14,15)(17,18,23,25)(19,26,20,22)(21,29,31,24)(27,30,28,32);(2,5)(6,7)(9,10,13,16)(11,14,12,15)(17,18,23,25)(19,26,20,22)(21,24,31,29)(27,32,28,30);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15)(17,19,23,20)(18,22,25,26)(21,27,31,28)(24,30,29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20,23,19)(18,22,25,26)(21,28,31,27)(24,30,29,32);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,15,16,14)(17,21,23,31)(18,24,25,29)(19,27,20,28)(22,30,26,32);(17,22)(18,19)(20,25)(21,30)(23,26)(24,27)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8094129);