#10097139

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

5348 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)(3,4)(5,8)(6,7)(17,18,23,22)(19,25,20,24)(21,26,29,30)(27,32,28,31);(1,3)(2,6)(4,8)(5,7)(9,10,13,16)(11,12,15,14)(17,19,23,20)(18,24,22,25)(21,27,29,28)(26,31,30,32);(1,4)(2,7)(3,8)(5,6)(9,11)(10,14)(12,16)(13,15);(2,8)(4,6)(9,12)(10,15)(11,16)(13,14)(17,20,23,19)(18,25,22,24)(21,27,29,28)(26,31,30,32);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,16)(13,14)(17,22,23,18)(19,25,20,24)(21,26,29,30)(27,31,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336552
maximal normal subgroup of	#72336552
maximal quotient of	#72336744

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10004335);