#6544078

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

3114 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)(23,30)(24,29)(25,33)(26,28)(27,31)(32,36)(34,35);(2,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23)(22,27)(24,31)(25,34)(26,32)(28,35)(29,36)(30,33);(6,10)(11,17)(12,18)(19,20)(21,23)(22,27)(24,31)(25,34)(26,32)(28,35)(29,36)(30,33);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(22,28)(27,35)(29,33)(30,36);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,24,26,25)(22,29,28,33)(23,31,32,34)(27,36,35,30)

Polycyclic presentation:

2^9|a^2=fgi,d^2=h,e^2=hi,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	#72283199, #72284171, #72284901
maximal normal subgroup of	#72283199, #72284171, #72284901

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451274);