#6435046

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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

3349 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6342242);