#418911

order	512
nilpotency class	3
isoclinism	isoclinic class of #418911
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15267
commutator, Frattini, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2479 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,h^c=hi,f^d=fi,g^d=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210507, #72237724, #72237729, …
maximal normal subgroup of	#72210507, #72237724, #72237729, …
maximal quotient of	#72240607, #72258156, C₂×#418911

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326107);