#202825

order	512
nilpotency class	3
isoclinism	isoclinic class of #19328
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

828 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)(5,6,7,9)(8,10,12,11)(14,19)(15,23)(16,17)(20,27)(21,22)(24,25);(1,3,4,2)(6,9)(10,11)(13,14,16,21,18,22,17,19)(15,20,25,28,24,27,23,26);(6,10)(8,12)(9,11)(13,15,16,25,18,24,17,23)(14,20,21,28,22,27,19,26);(6,9)(10,11)(13,16,18,17)(14,21,22,19)(15,23,24,25)(20,26,27,28)

Polycyclic presentation:

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

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, 110021);