#363914

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62814	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^b=fi,e^c=ei,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271110);