#6378171

order	512
nilpotency class	3
isoclinism	isoclinic class of #60771
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

1748 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)(16,31)(17,23)(18,24)(19,34)(20,25)(22,32)(26,41)(27,36)(28,37)(29,44)(30,38)(33,40)(35,42)(39,43);(5,6)(7,10)(8,9)(11,12)(13,15,20,30)(14,21,25,38)(16,27,17,26)(18,28,19,29)(22,39,40,35)(23,41,31,36)(24,37,34,44)(32,43,33,42);(6,9)(10,12)(13,16,20,17)(14,22,25,40)(15,26,30,27)(18,32,19,33)(21,35,38,39)(23,34,31,24)(28,42,29,43)(36,44,41,37);(5,7,8,11)(6,10,9,12)(13,17,20,16)(14,23,25,31)(15,27,30,26)(18,32,19,33)(21,36,38,41)(22,34,40,24)(28,42,29,43)(35,44,39,37);(1,3,4,2)(13,18,20,19)(14,24,25,34)(15,28,30,29)(16,32,17,33)(21,37,38,44)(22,23,40,31)(26,42,27,43)(35,36,39,41)

Polycyclic presentation:

2^9|a^2=h,b^2=i,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=cfi,d^b=di,g^c=gi,e^d=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, 6285367);