#219843

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	#2541	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1102 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)(7,8)(11,12)(13,14,17,20,16,18,23,25)(15,19,22,26,21,24,27,28);(1,2)(3,4)(6,11)(10,12)(14,18)(15,21)(20,25)(22,27);(5,6,9,11)(7,10,8,12)(13,15)(14,19)(16,21)(17,22)(18,24)(20,26)(23,27)(25,28);(2,4)(5,7,9,8)(6,10,11,12)(14,18)(15,21)(20,25)(22,27)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,c^b=chi,d^b=dei,d^c=di,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127039);