#10480832

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

5309 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,28)(19,26,30,23)(20,25,31,24)(21,29,32,27);(1,3,5,6)(2,4,7,8)(9,10,12,14)(11,13,16,15)(17,19)(18,23)(20,27)(21,24)(22,30)(25,32)(26,28)(29,31);(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,27)(21,23)(22,31)(25,28)(26,32)(29,30);(1,2)(3,8)(4,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,23)(22,31)(24,28)(26,32)(27,30);(1,2)(3,4)(5,7)(6,8)(17,19)(18,26)(20,27)(21,25)(22,30)(23,28)(24,32)(29,31);(1,4)(2,6)(3,7)(5,8)(17,21)(18,27)(19,24)(20,23)(22,32)(25,30)(26,31)(28,29)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300627
maximal normal subgroup of	#72300627
maximal quotient of	#72300630, #72300821

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10388028);