#6431725

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431723
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

7156 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:

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

Polycyclic presentation:

2^9|c^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,f^b=fi,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306978, #72307018, #72307021, …
maximal normal subgroup of	#72306978, #72307018, #72307021, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338921);