#276739

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

579 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=efh,c^2=i,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 183935);