#152876

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₁₆	C₂³
commutator, socle	C₂³	C₂²×C₁₆
prime radical	C₂²×D₄	C₁₆
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 330 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322704, #72322886, #72322888
maximal normal subgroup of	#72322704, #72322886, #72322888
maximal quotient of	#72322726

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60072);