#354788

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	C₄×#120
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3512 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,h^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	#72341715, #72341881, #72341891, …
maximal subgroup of	#72239887, #72257225, #72280138, …
maximal normal subgroup of	#72239887, #72257225, #72280138, …
maximal quotient of	#72239890, #72313241, #72319440, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261984);