#212924

order	512
nilpotency class	3
isoclinism	isoclinic class of #19745
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

840 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120120);