#6431889

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,h^b=hi,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306292, #72312244, #72312492, …
maximal normal subgroup of	#72306292, #72312244, #72312492, …
maximal quotient of	#72306329

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339085);