#6436537

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432833
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
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280979, #72282227, #72282286
maximal normal subgroup of	#72280979, #72282227, #72282286

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343733);