#354830

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
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	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214887, #72227309, #72239720, …
maximal normal subgroup of	#72214887, #72239720, #72250083, …
maximal quotient of	#72239742, #72308415, #72318965

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262026);