#420417

order	512
nilpotency class	3
isoclinism	isoclinic class of #419852
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1776 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255889, #72255903, #72255919
maximal normal subgroup of	#72255889, #72255903, #72255919

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327613);