#8201720

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

6266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249526, #72249536, #72249538
maximal normal subgroup of	#72249526, #72249536, #72249538

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8108916);