#119593

order	512
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26789);