#149371

order	512
nilpotency class	4
isoclinism	isoclinic class of #129554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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

526 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dehi,c^2=deh,d^2=hi,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=dh,e^b=eh,g^b=ghi,h^b=hi,d^c=dh,e^c=eh,g^c=ghi,h^c=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, 56567);