#382435

order	512
nilpotency class	3
isoclinism	isoclinic class of #213736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#3485	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252226, #72255699, #72258369, …
maximal normal subgroup of	#72252226, #72255699, #72258369, …
maximal quotient of	#72252222, #72258985, #72266193

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289631);