#8182104

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

5007 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=g,e^2=g,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, 8089300);