#285765

order	512
nilpotency class	3
isoclinism	isoclinic class of #281659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

638 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)(3,6)(4,5)(7,8)(9,10,12,16,13,11,14,15)(17,18,22,25,21,26,20,24)(19,27,30,32,28,23,29,31);(2,5)(6,8)(9,11,12,15,13,10,14,16)(17,19,22,30,21,28,20,29)(18,23,25,31,26,27,24,32);(1,3,4,7)(2,6,5,8)(10,15)(11,16)(12,14)(18,24)(19,28)(20,22)(23,31)(25,26)(27,32);(2,5)(6,8)(17,20,21,22)(18,25,26,24)(19,29,28,30)(23,31,27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=ei,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192961);