#296973

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

758 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)(9,10,14,16)(11,15,12,13)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(2,5)(6,7)(9,11,14,12)(10,13,16,15)(17,18,21,23)(19,22,26,29)(20,24,27,30)(25,28,31,32);(9,12,15,10,14,11,13,16)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(1,3,4,8)(2,6,5,7)(9,13,14,15)(10,12,16,11)(18,23)(19,26)(24,30)(25,31)

Polycyclic presentation:

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

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, 204169);