#354786

order	512
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3600 conjugacy classes of subgroups; only the 578 normal subgroups are shown.

normal subgroups only

This group has 3600 conjugacy classes of subgroups — too many to draw. Only its 578 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 — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72341974, #72342049, #72342063
maximal subgroup of	#72214895, #72281084, #72281292, …
maximal normal subgroup of	#72214895, #72281084, #72281292, …
maximal quotient of	#72281580, #72343175

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261982);