#301041

order	512
nilpotency class	3
isoclinism	isoclinic class of #264283
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×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1011 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)(5,6,10,12)(7,8,11,9)(13,14)(15,21)(16,25)(17,27)(18,20)(19,24)(22,26)(23,28);(5,7)(6,8)(9,12)(10,11)(13,15,17,24,18,23,26,16)(14,19,22,21,20,25,27,28);(1,2,3,4)(5,7,8,12,10,11,9,6)(14,20)(15,23)(16,24)(22,27);(5,8,10,9)(6,11,12,7)(13,16,26,23,18,24,17,15)(14,21,27,19,20,28,22,25)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=ehi,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=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 208237);