#205307

order	512
nilpotency class	3
isoclinism	isoclinic class of #19328
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1306 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)(5,6,9,11,10,7,8,12)(14,18)(15,21)(16,23)(19,26)(20,25)(22,27);(1,2,3,4)(5,7,10,6)(8,12,9,11)(13,14,16,18,17,20,23,25)(15,19,21,24,22,26,27,28);(13,15,16,21,17,22,23,27)(14,19,18,24,20,26,25,28);(5,8,10,9)(6,11,7,12)(14,20)(18,25)(19,26)(24,28)

Polycyclic presentation:

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

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, 112503);