#108401

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

530 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(2,3)(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20)(21,22)(23,27)(24,34,35,28)(25,30)(26,29)(31,43,44,38)(32,40)(33,39)(36,47,48,41)(37,42)(45,51,52,49)(46,50);(1,2,4,3)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,32,26,33,37,46)(22,27,30,40,29,39,42,50)(24,31,36,45,35,44,48,52)(28,38,41,49,34,43,47,51);(1,3,4,2)(7,13)(10,16)(14,19)(17,20)(21,24)(22,28)(23,31)(25,36)(26,35)(27,38)(29,34)(30,41)(32,45)(33,44)(37,48)(39,43)(40,49)(42,47)(46,52)(50,51)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,f^2=i,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=ci,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15597);