#242397

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149593);