#277443

order	512
nilpotency class	3
isoclinism	isoclinic class of #264203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

725 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ei,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 184639);