#6376433

order	512
nilpotency class	3
isoclinism	isoclinic class of #60768
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2374 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283629);