#408623

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#60453	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1294 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)(3,8)(4,6)(5,7)(9,10)(11,22)(12,26)(13,28)(14,21)(15,19)(16,20)(17,24)(18,31)(23,38)(25,34)(27,33)(29,35)(30,37)(32,36)(39,40);(9,11,15,25,16,24,14,12)(10,17,21,34,20,22,19,26)(13,23,31,29,32,39,30,27)(18,33,28,40,37,35,36,38);(1,3,5,4)(2,6,7,8)(9,12,14,24,16,25,15,11)(10,17,19,26,20,22,21,34)(13,29,30,23,32,27,31,39)(18,35,36,40,37,33,28,38);(1,4,5,3)(2,6,7,8)(9,13,14,30,16,32,15,31)(10,18,19,36,20,37,21,28)(11,23,12,27,24,39,25,29)(17,33,26,38,22,35,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307003, #72307302, #72307328
maximal normal subgroup of	#72307003, #72307302, #72307328

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315819);