#109563

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₄×C₄⋊C₄
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=ci,d^b=dh,e^b=eh,d^c=dh,e^c=eh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240050
maximal normal subgroup of	#72240050
maximal quotient of	#72240056, #72240057

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16759);