#492886

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂²×D₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2426 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,b^a=be,e^a=eh,h^a=hi,c^b=ci,f^b=fh,f^e=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256629, #72256630, #72256633, …
maximal normal subgroup of	#72256629, #72256630, #72256633, …
maximal quotient of	#72263740, #72303608, #72313906, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400082);