#242464

order	512
nilpotency class	3
isoclinism	isoclinic class of #233080
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

872 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=h,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,d^b=dh,d^c=dh,e^c=eh,f^c=fi,g^c=gh

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, 149660);