#6380651

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,f^b=fi,h^b=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287847);