#146422

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×#107
prime radical	C₂⁵	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

990 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 290 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)(3,4)(5,6)(7,8)(9,10,13,18,15,21,27,20)(11,22,24,36,26,37,38,31)(12,19,14,17,28,34,29,35)(16,25,30,39,33,40,32,23);(2,4)(9,11,14,25,13,24,28,39,15,26,29,40,27,38,12,23)(10,16,19,31,20,32,35,37,21,33,34,36,18,30,17,22);(6,8)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287528, #72287546, #72287553
maximal normal subgroup of	#72287528, #72287546, #72287553

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53618);