#181011

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1968 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88207);