#370107

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=g,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321897, #72342922, #72342923, …
maximal normal subgroup of	#72321897, #72342922, #72342923, …
maximal quotient of	#72343021

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277303);