#8178765

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

5511 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:

(1,2,5,6)(3,7,8,4)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32);(1,3)(2,4)(5,8)(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);(2,6)(4,7)(10,13)(14,15)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32);(1,3)(2,4)(5,8)(6,7)(17,21,23,30)(18,25,22,29)(19,27,20,28)(24,31,26,32);(1,4)(2,3)(5,7)(6,8)(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=gh,c^2=gh,d^2=gh,e^2=gh,f^2=gh,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, 8085961);