#6441746

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2261 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,9)(8,10)(11,12)(13,14,18,25,20,24,19,26)(15,27,30,38,32,39,31,21)(16,22,17,23,34,40,33,41)(28,43,44,42,35,37,29,36);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,21)(16,29)(17,35)(18,31)(19,30)(20,32)(22,37)(23,42)(24,38)(25,39)(26,27)(28,33)(34,44)(36,41)(40,43);(2,4)(6,10)(9,12)(13,16,19,33,20,34,18,17)(14,22,26,41,24,40,25,23)(15,28,31,29,32,35,30,44)(21,36,39,37,38,42,27,43);(2,4)(13,17,18,34,20,33,19,16)(14,23,25,40,24,41,26,22)(15,29,30,28,32,44,31,35)(21,37,27,36,38,43,39,42);(14,24)(21,38)(22,40)(23,41)(25,26)(27,39)(36,42)(37,43)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348942);