#452204

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1126 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:

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

Polycyclic presentation:

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

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, 359400);