#236196

order	512
nilpotency class	3
isoclinism	isoclinic class of #234079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1408 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=dhi,e^b=eh,f^b=fi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143392);