#110056

order	512
nilpotency class	3
isoclinism	isoclinic class of #1909
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

763 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,7,6,8,5,3)(9,10)(11,19,22,15)(12,23)(13,26)(14,17)(16,25)(18,28)(20,34,35,40)(21,30,27,32)(24,37)(29,36,33,39)(31,38);(1,3,5,8,6,7,4,2)(9,11)(10,15)(12,20)(13,27)(14,22)(16,29)(17,19)(18,32)(21,28)(23,34)(24,35)(25,39)(26,30)(31,33)(36,38)(37,40);(1,2,4,7,6,8,5,3)(9,12,13,25,14,24,28,38)(10,16,18,23,17,31,26,37)(11,20,27,39,22,35,21,36)(15,29,32,34,19,33,30,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,e^b=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, 17252);