#6399564

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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	C₂×#3269	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2195 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)(5,6)(7,10)(8,9)(11,12)(13,14,20,25)(15,21,30,38)(16,31,17,23)(18,24,19,34)(22,33,39,32)(26,40,27,36)(28,37,29,43)(35,42,44,41);(6,9)(7,11)(13,15,20,30)(14,21,25,38)(16,26,17,27)(18,28,19,29)(22,35,39,44)(23,36,31,40)(24,37,34,43)(32,41,33,42);(13,16,20,17)(14,22,25,39)(15,26,30,27)(18,32,19,33)(21,35,38,44)(23,34,31,24)(28,41,29,42)(36,43,40,37);(5,7,8,11)(6,10,9,12)(13,17,20,16)(14,23,25,31)(15,27,30,26)(18,32,19,33)(21,36,38,40)(22,34,39,24)(28,41,29,42)(35,43,44,37);(1,3,4,2)(6,9)(10,12)(13,18,20,19)(14,24,25,34)(15,28,30,29)(16,32,17,33)(21,37,38,43)(22,23,39,31)(26,41,27,42)(35,36,44,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,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, 6306760);