#6533350

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283032, #72283872, #72284239
maximal normal subgroup of	#72283032, #72283872, #72284239

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6440546);