#6463028

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	#60639	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370224);