#213214

order	512
nilpotency class	3
isoclinism	isoclinic class of #19671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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

826 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)(6,9)(7,11)(10,12)(13,14,16,18)(15,21,23,28)(17,25,22,20)(19,27,26,24);(5,6,7,9,8,10,11,12)(14,18)(19,26)(20,25)(21,28);(6,10)(9,12)(13,15,22,27,16,23,17,24)(14,19,25,28,18,26,20,21);(1,3,4,2)(15,23)(19,26)(21,28)(24,27)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fi,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,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, 120410);