#365125

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂×#585	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1903 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,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	#72321017, #72342923, #72342934, …
maximal normal subgroup of	#72321017, #72342923, #72342934, …
maximal quotient of	#72343055

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272321);