#242276

order	512
nilpotency class	3
isoclinism	isoclinic class of #233958
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

889 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)(11,16)(12,15)(13,14)(17,18,21,24)(19,26,27,20)(22,31,30,25)(23,32,28,29);(2,5)(6,8)(9,11,12,13)(10,14,15,16)(17,19,21,27)(18,20,24,26)(22,23,30,28)(25,32,31,29);(10,14)(11,13)(15,16)(17,20,21,26)(18,23,24,28)(19,25,27,31)(22,32,30,29);(1,3,4,7)(2,6,5,8)(19,27)(20,26)(23,28)(29,32)

Polycyclic presentation:

2^9|a^2=i,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,d^b=di,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, 149472);