#419028

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2235 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,10,14,11,16,9,15)(7,17,21,29,22,19,20,12)(8,23,26,32,27,31,25,13)(18,28,24,33,35,36,34,30);(1,2,3,4)(5,7,9,20,11,22,10,21)(6,12,14,17,16,29,15,19)(8,24,25,18,27,34,26,35)(13,30,23,28,32,33,31,36);(5,8)(6,13)(7,18)(9,26)(10,25)(11,27)(12,28)(14,31)(15,23)(16,32)(17,30)(19,33)(20,24)(21,34)(22,35)(29,36);(2,4)(5,6)(7,19)(8,13)(9,15)(10,14)(11,16)(12,21)(17,22)(18,33)(20,29)(23,26)(24,36)(25,31)(27,32)(28,34)(30,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250092, #72252967, #72256873
maximal normal subgroup of	#72250092, #72252967, #72256873

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326224);