#194484

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1172 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101680);