#6444916

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	#24266	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2067 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 6352112);