#443518

order	512
nilpotency class	3
isoclinism	isoclinic class of #417460
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1638 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1638 conjugacy classes of subgroups — too many to draw. Only its 266 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=g,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi

maximal subgroup of	#72226903, #72226934, #72226944
maximal normal subgroup of	#72226903, #72226934, #72226944

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 350714);