#183699

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₄⋊C₄
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90895);