#115461

order	512
nilpotency class	3
isoclinism	isoclinic class of #10120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

952 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)(6,9)(7,11)(10,12)(13,14)(15,24)(16,29)(17,34)(18,22)(19,23)(20,26)(21,32)(25,42)(27,37)(28,36)(30,41)(31,40)(33,39)(35,43)(38,44);(7,11)(9,12)(13,15)(14,20)(16,30)(17,27)(18,26)(19,28)(21,38)(22,36)(23,37)(24,34)(25,31)(29,35)(32,43)(33,44)(39,42)(40,41);(5,6)(7,12)(8,10)(9,11)(13,16,19,33)(14,21,23,40)(15,25,28,43)(17,31,18,32)(20,35,37,42)(22,39,34,29)(24,41,36,38)(26,30,27,44)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ehi,f^c=fh

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, 22657);