#270990

order	512
nilpotency class	3
isoclinism	isoclinic class of #270875
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1114 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 178186);