#126910

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

950 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,e^2=h,b^a=bd,d^a=dg,g^a=gi,c^b=ci,e^b=eg,e^d=ei

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, 34106);