#6384476

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2029 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291672);