#149452

order	512
nilpotency class	4
isoclinism	isoclinic class of #129554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(13,14,18,22,17,21,32,23)(15,24,27,37,25,40,33,41)(16,28,31,20,30,39,44,38)(19,29,35,26,34,43,36,42);(5,6,8,10)(7,11,12,9)(13,15,17,25)(14,19,21,34)(16,29,30,43)(18,33,32,27)(20,37,38,41)(22,36,23,35)(24,28,40,39)(26,31,42,44);(2,4)(5,7,8,12)(6,9,10,11)(15,25)(19,34)(24,40)(26,42)(27,33)(29,43)(35,36)(37,41)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,d^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dhi,g^b=gi,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56648);