#6533284

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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3656 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)(15,24)(16,22)(17,23)(18,21)(19,26)(20,25)(27,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,25)(17,26)(18,27)(20,21)(22,24)(23,28);(6,9)(10,12)(13,15)(14,20)(16,25)(17,26)(18,27)(19,21)(22,28)(23,24);(5,7)(6,10)(8,11)(9,12)(14,21)(19,20)(22,23)(24,28);(7,11)(10,12)(13,16,18,17)(14,22,21,23)(15,25,27,26)(19,24,20,28)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283671, #72284026, #72284901
maximal normal subgroup of	#72283671, #72284026, #72284901

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6440480);