#130001

order	512
nilpotency class	4
isoclinism	isoclinic class of #129769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

998 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,h^b=hi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37197);