#6529186

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308175, #72308783, #72308811
maximal normal subgroup of	#72308175, #72308783, #72308811

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6436382);