#411604

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(10,12)(13,14,19,24)(15,25,27,36)(16,30,33,37)(17,23,18,22)(20,29,35,28)(21,31,38,32)(26,40,41,34)(39,42,44,43);(2,4)(5,6)(7,8)(13,15,19,27)(14,20,24,35)(16,26,33,41)(17,29,18,28)(21,34,38,40)(22,36,23,25)(30,44,37,39)(31,43,32,42);(6,8)(13,16,19,33)(14,21,24,38)(15,26,27,41)(17,32,18,31)(20,34,35,40)(22,30,23,37)(25,39,36,44)(28,43,29,42);(9,10,11,12)(15,27)(20,35)(25,36)(26,41)(28,29)(34,40)(39,44)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318800);