#413174

order	512
nilpotency class	3
isoclinism	isoclinic class of #395151
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3476 conjugacy classes of subgroups; only the 410 normal subgroups are shown.

normal subgroups only

This group has 3476 conjugacy classes of subgroups — too many to draw. Only its 410 normal subgroups are shown.

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

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 20 points:

(2,4)(6,8)(14,15)(16,17)(19,20);(1,2,3,4)(9,10)(11,12)(13,14,17,19,18,20,16,15);(5,6,7,8)(10,12);(13,15,16,20,18,19,17,14)

Polycyclic presentation:

2^9|b^2=ei,c^2=f,d^2=hi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg

maximal subgroup of	#72280207, #72280277
maximal normal subgroup of	#72280207, #72280277

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320370);