#6344602

order	512
nilpotency class	3
isoclinism	isoclinic class of #60460
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂×#578	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1739 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,f^2=h,g^2=i,b^a=bf,c^a=chi,f^a=fh,c^b=ci,f^b=fh,e^d=eh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322724, #72322739, #72322801
maximal normal subgroup of	#72322724, #72322739, #72322801

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251798);