#242518

order	512
nilpotency class	3
isoclinism	isoclinic class of #233019
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

840 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=hi,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=dhi,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, 149714);