#272865

order	512
nilpotency class	3
isoclinism	isoclinic class of #272795
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	#37770	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1123 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,e^c=ehi,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, 180061);