#152829

order	512
nilpotency class	5
isoclinism	isoclinic class of #151115
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2177
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

219 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 219 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=dh,d^2=i,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^b=ei,d^c=di,e^c=egi,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225508, #72319601, #72319630
maximal normal subgroup of	#72225508, #72319601, #72319630
maximal quotient of	#72225503, #72225531

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60025);