#366315

order	512
nilpotency class	3
isoclinism	isoclinic class of #23900
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

1914 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342922, #72342934, #72342939
maximal normal subgroup of	#72342922, #72342934, #72342939
maximal quotient of	#72343038

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273511);