#8689879

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

5750 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)(5,7)(6,8)(18,22)(19,25)(24,29)(26,31);(1,3,5,8)(2,6,7,4)(9,10)(11,15)(12,14)(13,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(2,7)(3,8)(10,14)(13,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,16)(13,14)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326661
maximal normal subgroup of	#72326661
maximal quotient of	#72326697, #72326712, #72326989, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8597075);