#242004

order	512
nilpotency class	3
isoclinism	isoclinic class of #233936
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

803 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149200);