#152500

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#2215	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

398 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 110 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)(5,6)(7,8)(9,10,11,16,21,29,36,27,13,18,22,30,25,33,37,28,14,19,23,31,38,40,39,34,24,32,35,26,12,17,20,15);(1,2,3,4)(10,15)(11,20)(12,21)(13,24)(16,17)(18,34)(19,28)(22,39)(23,37)(25,38)(26,29)(27,32)(30,40)(31,33)(35,36);(6,8)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gh,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^b=dg,f^b=fg,g^b=gh,h^b=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286704, #72322872, #72322888
maximal normal subgroup of	#72286704, #72322872, #72322888
maximal quotient of	#72286857, #72321629

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59696);