#129397

order	512
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2274 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 654 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36593);