#495073

order	512
nilpotency class	4
isoclinism	isoclinic class of #483425
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

674 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237237, #72239487, #72239538, …
maximal normal subgroup of	#72237237, #72239487, #72239538, …
maximal quotient of	#72319184, #72320101, C₂×#495073

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402269);