#6431724

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

7228 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)(7,15)(8,12)(9,21)(10,14)(11,17)(13,23)(16,29)(18,28)(19,26)(20,33)(22,24)(25,31)(27,32)(30,36)(34,35);(5,7,10,18)(6,11,14,24)(8,16,19,30)(9,22,23,17)(12,25,26,35)(13,28,21,15)(20,34,32,31)(27,36,33,29);(2,4)(9,23)(11,24)(13,21)(17,22)(20,32)(25,35)(27,33)(31,34);(5,8)(6,12)(7,16)(9,20)(10,19)(11,25)(13,27)(14,26)(15,29)(17,31)(18,30)(21,33)(22,34)(23,32)(24,35)(28,36);(8,19)(12,26)(16,30)(20,32)(25,35)(27,33)(29,36)(31,34)

Polycyclic presentation:

2^9|b^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	#72286244, #72287334, #72287345, …
maximal normal subgroup of	#72286244, #72287334, #72287345, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338920);