#8186300

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

5733 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)(3,5)(4,6)(7,8)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32);(2,3)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18,23,22)(19,26,20,24)(21,25,30,29)(27,32,28,31);(4,8)(6,7)(10,13)(14,15)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,14,13,15)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32);(17,21,23,30)(18,25,22,29)(19,27,20,28)(24,31,26,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,22,23,18)(19,24,20,26)(21,29,30,25)(27,31,28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gi,e^2=h,f^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, 8093496);