#299115

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1108 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:

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

Polycyclic presentation:

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

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, 206311);