#8388328

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

4960 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8295524);