#6382200

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6289396);