#152828

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	C₂²×D₁₆	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

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

normal subgroups only

This group has 228 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)(6,10)(7,17,19,35)(8,11)(9,22)(13,28,31,36)(14,21)(15,16)(18,32,20,34)(23,25)(24,27,30,29)(26,33);(1,2,4,3)(5,6,11,15,12,16,8,10)(7,13,18,29,19,31,20,27)(9,23,25,22,26,14,21,33)(17,24,34,28,35,30,32,36);(1,3,4,2)(5,7)(6,13)(8,20)(9,24)(10,27)(11,18)(12,19)(14,32)(15,29)(16,31)(17,33)(21,36)(22,35)(23,34)(25,28)(26,30)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=dh,d^2=i,e^2=ghi,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	#72319425, #72319593, #72319601
maximal normal subgroup of	#72319425, #72319593, #72319601

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60024);