#268532

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
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	#59034	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1031 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175728);