#118450

order	512
nilpotency class	3
isoclinism	isoclinic class of #10464
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
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

720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25646);