#436314

order	512
nilpotency class	3
isoclinism	isoclinic class of #421903
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

842 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=e,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 343510);