#117815

order	512
nilpotency class	3
isoclinism	isoclinic class of #10120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=eh,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25011);