#130823

order	512
nilpotency class	4
isoclinism	isoclinic class of #130512
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2351 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

603 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,f^b=fgh,f^c=fgi,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38019);