#402828

order	512
nilpotency class	3
isoclinism	isoclinic class of #396113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=ef,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^d=hi

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, 310024);