#6378162

order	512
nilpotency class	3
isoclinism	isoclinic class of #60768
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 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)(7,11)(13,14)(15,21)(16,30)(17,23)(18,24)(19,34)(20,25)(22,31)(26,36)(27,37)(28,43)(29,38)(32,40)(33,39)(35,41)(42,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,26)(17,32)(18,27)(19,28)(20,29)(22,35)(23,40)(24,37)(25,38)(30,36)(31,41)(33,42)(34,43)(39,44);(6,9)(10,12)(13,16,20,17)(14,22,25,39)(15,26,29,32)(18,31,19,33)(21,35,38,44)(23,34,30,24)(27,41,28,42)(36,37,40,43);(5,7,8,11)(6,10,9,12)(13,17,20,16)(14,23,25,30)(15,26,29,32)(18,31,19,33)(21,36,38,40)(22,34,39,24)(27,42,28,41)(35,37,44,43);(1,3,4,2)(13,18,20,19)(14,24,25,34)(15,27,29,28)(16,31,17,33)(21,37,38,43)(22,23,39,30)(26,41,32,42)(35,40,44,36)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,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, 6285358);