#273020

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
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	C₂³×C₄	C₂⁴
prime radical	#2617	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1063 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180216);