#6344477

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
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	C₂³×C₄⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1881 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322940, #72322956, #72322974
maximal normal subgroup of	#72322940, #72322956, #72322974

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251673);