#181028

order	512
nilpotency class	3
isoclinism	isoclinic class of #17392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1838 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,f^2=i,b^a=be,d^a=di,e^a=eh,c^b=ci,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88224);