#6442609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437473
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4650 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287151, #72287261, #72287270, …
maximal normal subgroup of	#72287151, #72287261, #72287270, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349805);