#411941

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=h,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,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264600, #72264612, #72307937
maximal normal subgroup of	#72264600, #72264612, #72307937

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319137);