#6376443

order	512
nilpotency class	3
isoclinism	isoclinic class of #60753
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2434 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)(15,23)(16,27)(17,21)(18,32)(19,25)(20,31)(22,29)(24,38)(26,34)(28,42)(30,41)(33,43)(35,40)(36,44)(37,39);(1,3,4,2)(5,7,10,6)(8,11,9,12)(13,15)(14,19)(16,28)(17,25)(18,26)(20,35)(21,23)(22,34)(24,30)(27,38)(29,39)(31,43)(32,37)(33,36)(40,44)(41,42);(5,8,10,9)(6,11,7,12)(13,16,29,44,17,30,18,31)(14,20,32,41,21,36,22,27)(15,24,39,43,25,28,26,40)(19,33,37,38,23,35,34,42);(5,9,10,8)(6,11,7,12)(14,21)(15,25)(20,36)(22,32)(24,28)(26,39)(27,41)(40,43);(16,30)(20,36)(24,28)(27,41)(31,44)(33,35)(38,42)(40,43)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283639);