#300853

order	512
nilpotency class	3
isoclinism	isoclinic class of #264203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1081 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:

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

Polycyclic presentation:

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

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, 208049);