#117424

order	512
nilpotency class	3
isoclinism	isoclinic class of #10128
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24620);