#6544255

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3135 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)(21,22,26,34,25,30,24,29)(23,28,31,27,33,35,32,36);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23)(22,27)(24,31)(25,33)(26,32)(28,34)(29,35)(30,36);(2,4)(6,10)(11,17)(12,18)(19,20)(21,23)(22,28)(24,32)(25,33)(26,31)(27,34)(29,36)(30,35);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,24,25,26)(22,29,30,34)(23,31,33,32)(27,35,36,28);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(22,30)(27,36)(28,35)(29,34)

Polycyclic presentation:

2^9|a^2=fgi,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,f^b=fi,d^c=di,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308249, #72308348, #72308427
maximal normal subgroup of	#72308249, #72308348, #72308427

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451451);