#458698

order	512
nilpotency class	3
isoclinism	isoclinic class of #453991
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
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

1004 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)(9,10,13,17,14,18,27,34)(11,19,21,29,22,32,25,15)(12,16,23,30,24,31,26,33)(20,35,37,40,38,36,39,28);(1,3,5,4)(2,6,7,8)(9,11,14,22)(10,15,18,29)(12,20,24,38)(13,25,27,21)(16,28,31,40)(17,32,34,19)(23,39,26,37)(30,36,33,35);(1,4,5,3)(2,6,7,8)(9,12,13,23,14,24,27,26)(10,16,17,30,18,31,34,33)(11,20,21,37,22,38,25,39)(15,28,19,35,29,40,32,36);(9,12)(10,16)(11,20)(13,26)(14,24)(15,28)(17,33)(18,31)(19,36)(21,39)(22,38)(23,27)(25,37)(29,40)(30,34)(32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365894);